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Grain Boundary Properties: Energy, Mobility

Grain Boundary Properties: Energy, Mobility. 27-765, Spring 2001 A.D. Rollett. Why learn about grain boundary properties?. Many aspects of materials behavior and performance affected by g.b. properties.

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Grain Boundary Properties: Energy, Mobility

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  1. Grain Boundary Properties:Energy, Mobility 27-765, Spring 2001 A.D. Rollett G.B. properties

  2. Why learn about grain boundary properties? • Many aspects of materials behavior and performance affected by g.b. properties. • Examples include:- stress corrosion cracking in Pb battery electrodes, Ni-alloy nuclear fuel containment, steam generator tubes- creep strength in high temp. alloys- weld cracking (under investigation)- electromigration resistance (interconnects) G.B. properties

  3. Properties, phenomena of interest 1. Energy (excess free energy  wetting, precipitation) 2. Mobility (normal motion  grain growth, recrystallization) 3. Sliding (tangential motion  creep) 4. Cracking resistance (intergranular fracture) 5. Segregation of impurities (embrittlement, formation of second phases) G.B. properties

  4. 1. Grain Boundary Energy • First categorization of boundary type is into low-angle versus high-angle boundaries. Typical value in cubic materials is 15° for the misorientation angle. • Read-Shockley model describes the energy variation with angle successfully in many experimental cases, based on a dislocation structure. G.B. properties

  5. LAGB to HAGB Transition • LAGB: steep risewith angle.HAGB: plateau Disordered Structure Dislocation Structure G.B. properties

  6. 1.1 Read-Shockley model • Start with a symmetric tilt boundary composed of a wall of infinitely straight, parallel edge dislocations (e.g. based on a 100, 111 or 110 rotation axis with the planes symmetrically disposed). • Dislocation density (L-1) given by:1/D = 2sin(q/2)/b  q/b for small angles. G.B. properties

  7. b 1.1 Tilt boundary D G.B. properties

  8. 1.1 Read-Shockley contd. • For an infinite array of edge dislocations the long-range stress field depends on the spacing. Therefore given the dislocation density and the core energy of the dislocations, the energy of the wall (boundary) is estimated (r0 sets the core energy of the dislocation):ggb = E0 q(A0 - lnq), whereE0 = µb/4π(1-n); A0 = 1 + ln(b/2πr0) G.B. properties

  9. 1.1 LAGB experimental results • Experimental results on copper. [Gjostein & Rhines, Acta metall. 7, 319 (1959)] G.B. properties

  10. 1.1 Read-Shockley contd. • If the non-linear form for the dislocation spacing is used, we obtain a sine-law variation (Ucore= core energy):ggb = sin|q| {Ucore/b - µb2/4π(1-n)ln(sin|q|)} • Note: this form of energy variation may also be applied to CSL-vicinal boundaries. G.B. properties

  11. Yang, C.-C., A. D. Rollett, et al. (2001). “Measuring relative grain boundary energies and mobilities in an aluminum foil from triple junction geometry.” Scripta Materiala: in press. [001] 0.33 0.30 0.26 0.23 [111] [101] Low Angle Grain Boundary Energy High [117] [105] [113] [205] [215] [335] [203] Low [8411] [323] [727] A. Otsuki, Ph.D.thesis, Kyoto University, Japan (1990)  vs. G.B. properties

  12. 1.2 Energy of High Angle Boundaries • No universal theory exists to describe the energy of HAGBs. • Abundant experimental evidence for special boundaries at (a small number) of certain orientations. • Each special point (in misorientation space) expected to have a cusp in energy, similar to zero-boundary case but with non-zero energy at the bottom of the cusp. G.B. properties

  13. 1.2 Exptl. Observations <100>Tilts Twin <110>Tilts G.B. properties Hasson, G. C. and C. Goux (1971). “Interfacial energies of tilt boundaries in aluminum. Experimental and theoretical determination.” Scripta metallurgica5: 889-894

  14. Dislocation models of HAGBs • Boundaries near CSL points expected to exhibit dislocation networks, which is observed. <100> twists G.B. properties Howe, J. M. (1997). Interfaces in Materials. New York, Wiley Interscience.

  15. 1.2 Atomistic modeling • Extensive atomistic modeling has been conducted using (mostly) embedded atom potentials and an energy-relaxation method to locate the minimum energy configuration of a (finite) bicrystal. See Wolf & Yip, Materials Interfaces: Atomic-Level Structure & Properties, Chapman & Hall, 1992; also book by Sutton & Balluffi. • Grain boundaries in fcc metals: Cu, Au G.B. properties

  16. Atomistic models: results • Results of atomistic modeling confirm the importance of the more symmetric boundaries. G.B. properties

  17. Coordination Number Reasonable correlation for energy versus the coordination number for atoms at the boundary: suggests that broken bond model may be applicable, as it is for solid/vapor surfaces. G.B. properties

  18. Experimental Impact of Energy • Wetting by liquids is sensitive to grain boundary energy. • Example: copper wets boundaries in iron at high temperatures. • Wet versus unwetted condition found to be sensitive to grain boundary energy in Fe+Cu system: Takashima, M., A. D. Rollett, et al. (1999). Correlation of grain boundary character with wetting behavior. ICOTOM-12, Montréal, Canada, NRC Research Press, p.1647. G.B. properties

  19. G.B. Energy: Metals: Summary • For low angle boundaries, use the Read-Shockley model: well established both experimentally and theoretically. • For high angle boundaries, use a constant value unless near a CSL structure with high fraction of coincident sites and plane suitable for good atomic fit. G.B. properties

  20. LA->HAGB Transition High Angle Boundaries Transfer of atoms from the shrinking grain to the growing grain by atomic bulk diffusion mechanism Low Angle Boundaries Transfer of vacancies between two adjacent sets of dislocations by grain boundary diffusion mechanism G.B. properties

  21. 2.1 Low Angle G.B. Mobility • Mobility of low angle boundaries dominated by climb of the dislocations making up the boundary. • Even in a symmetrical tilt boundary the dislocations must move non-conservatively in order to maintain the correct spacing as the boundary moves. G.B. properties

  22. boundary displacement  h dx Tilt Boundary Motion Burgers vectors inclined with respect to the boundary plane in proportion to the misorientation angle. climb glide (Bauer and Lanxner, Proc. JIMIS-4 (1986) 411) G.B. properties

  23. Low Angle GB Mobility • Huang and Humphreys (2000): coarsening kinetics of subgrain structures in deformed Al single crystals. Dependence of the mobility on misorientation was fitted with a power-law relationship, M*=kqc, with c~5.2 and k=3.10-6 m4(Js)-1. • Yang, et al.: mobility (and energy) of LAGBs in aluminum: strong dependence of mobility on misorientation; boundaries based on [001] rotation axes had much lower mobilities than either [110] or [111] axes. G.B. properties

  24. Relative Mobility 0.9 0.3 0.1 0.03 0.01 0.0004 LAGB Mobility in Al, experimental [001] Low [117] [105] [113] [205] [215] [335] [203] [8411] [111] High [101] [323] [727] M vs. G.B. properties

  25. LAGB: Axis Dependence • We can explain the (strong) variation in LAGB mobility from <111> axes to <100> axes, based on the simple tilt model: <111> tilt boundaries have dislocations with Burgers vectors nearly perp. to the plane. <100> boundaries, however, have Burgers vectors near 45° to the plane. Therefore latter require more climb for a given displacement of the boundary. G.B. properties

  26. Symmetrical <001> 11.4o grain boundary=> nearly 45o alignment of dislocations with respect to the boundary normal =>  = 45o +/2 Symmetrical <111> 12.4o grain boundary=> dislocations are nearly parallel to the boundary normal =>  = /2 G.B. properties

  27. 2.1 Low Angle GB Mobility, contd. • Winning et al. Measured mobilities of low angle grain <112> and <111> tilt boundaries under a shear stress driving force. A sharp transition in activation enthalpy from high to low with increasing misorientation (at ~ 13°). G.B. properties

  28. Dislocation Modelsfor Low Angle G.B.s Sutton and Balluffi (1995). Interfaces in Crystalline Materials. Clarendon Press, Oxford, UK. G.B. properties

  29. Theory: Diffusion • Atom flux, J, between the dislocations is:where DL is the atom diffusivity (vacancy mechanism) in the lattice;m is the chemical potential;kT is the thermal energy;and Wis an atomic volume. G.B. properties

  30. Driving Force • A stress t that tends to move dislocations with Burgers vectors perpendicular to the boundary plane, produces a chemical potential gradient between adjacent dislocations associated with the non-perpendicular component of the Burgers vector: where dis the distance between dislocations in the tilt boundary. G.B. properties

  31. Atom Flux • The atom flux between the dislocations (per length of boundary in direction parallel to the tilt axis) passes through some area of the matrix between the dislocations which is very roughly A≈d/2. The total current of atoms between the two adjacent dislocations (per length of boundary) I is [SB]. G.B. properties

  32. Dislocation Velocity • Assuming that the rate of boundary migration is controlled by how fast the dislocations climb, the boundary velocity can be written as the current of atoms to the dislocations (per length of boundary in the direction parallel to the tilt axis) times the distance advanced per dislocation for each atom that arrives times the unit length of the boundary. G.B. properties

  33. Mobility (Lattice Diffusion only) • The driving force or pressure on the boundary is the product of the Peach-Koehler force on each dislocation times the number of dislocations per unit length, (since d=b/√2q). • Hence, the boundary mobility is [SB]:See also: Furu and Nes (1995), Subgrain growth in heavily deformed aluminium - experimental investigation and modelling treatment. Acta metall. mater., 43, 2209-2232. G.B. properties

  34. Theory: Addition of a Pipe Diffusion Model • Consider a grain boundary containing two arrays of dislocations, one parallel to the tilt axis and one perpendicular to it. Dislocations parallel to the tilt axis must undergo diffusional climb, while the orthogonal set of dislocations requires no climb. The flux along the dislocation lines is: G.B. properties

  35. Lattice+Pipe Diffusion • The total current of atoms from one dislocation parallel to the tilt axis to the next (per length of boundary) is where d is the radius of the fast diffusion pipe at the dislocation core and d1 and d2 are the spacing between the dislocations that run parallel and perpendicular to the tilt axis, respectively. G.B. properties

  36. Boundary Velocity • The boundary velocity is related to the diffusional current as above but with contributions from both lattice and pipe diffusion: G.B. properties

  37. Mobility (Lattice and Pipe Diffusion) • The mobility M=v/(tq) is now simply:This expression suggests that the mobility increases as the spacing between dislocations perpendicular to the tilt axis decreases. G.B. properties

  38. Effect of twist angle • If the density of dislocations running perpendicular to the tilt axis is associated with a twist component, then:where f is the twist misorientation. On the other hand, a network of dislocations with line directions running both parallel and perpendicular to the tilt axis may be present even in a pure tilt boundary assuming that dislocation reactions occur. G.B. properties

  39. Effect of Misorientation • If the density of the perpendicular dislocations is proportional to the density of parallel ones, then the mobility is:where a is a proportionality factor. Note the combination of mobility increasing and decreasing with misorientation. G.B. properties

  40. Results: Ni Mobility • Nickel: QL=2.86 eV, Q=0.6QL, D0L=D0=10-4 m2/s, b=3x10-10 m, W=b3, d=b, a=1, k=8.6171x10-5 eV/K. M (10-10 m4/[J s]) T (˚K) q (˚) G.B. properties

  41. Theory: Reduced Mobility • Product of the two quantities M*=Mg that is typically determined when g.b. energy not measured. Using the Read-Shockley expression for the grain boundary energy, we can write the reduced mobility as: G.B. properties

  42. Results: Ni Reduced Mobility • g0=1 J/m2 and q*=25˚, corresponding to a maximum in the boundary mobility at 9.2˚. log10M* (10-11m2/s) q (˚) T (˚K) G.B. properties

  43. Results: AluminumMobility vs. T and q The vertical axis is Log10 M. log10M (µm4/s MPa) g0 = 324 mJ/m2, q*= 15°, DL(T) 1.76.10-5 exp-{126153 J/mol/RT} m2/s, D(T) 2.8.10-6 exp-{81855 J/mol/RT} m2/s, d=b, b = 0.286 nm, W = 16.5.10-30m3 = b3/√2, a = 1. q (˚) T (K) G.B. properties

  44. Comparison with Expt.: Mobility vs. Angle at 873K Log10M (µm4/s MPa) 0 -1-2 -3 -4 -5 Log10M (µm4/s MPa) q (˚) M. Winning, G. Gottstein & L.S. Shvindlerman, Grain Boundary Dynamics under the Influence of MechanicalStresses, Risø-21 “Recrystallization”, p.645, 2000. G.B. properties

  45. Comparison with Expt.: Mobility vs. Angle at 473K Log10M (µm4/s MPa) Log10M (µm4/s MPa) 4 32 1 q (˚) G.B. properties

  46. Discussion on LAGB mobility • The experimental data shows high and low angle plateaus: the theoretical results are much more continuous. • The low T minimum is quite sharp compared with experiment. • Simple assumptions about the boundary structure do not capture the real situation. G.B. properties

  47. 2.1 LAGB mobility; conclusion • Agreement between calculated (reduced) mobility and experimental results is remarkably good. Only one (structure sensitive) adjustable parameter (a = 1), which determines the position of the minimum. • Better models of g.b. structure will permit prediction of low angle g.b. mobilities for all crystallographic types. G.B. properties

  48. LAGB to HAGB Transitions • Read-Shockley forenergy of low angleboundaries • Exponentialfunction for transitionfrom low- to high-angle boundaries G.B. properties

  49. High Angle GB Mobility • Large variations known in HAGB mobility. • Classic example is the high mobility of boundaries close to 40°<111> (which is near the S7 CSL type). • Note broad maximum. Gottstein & Shvindlerman: grain boundary migration in metals G.B. properties

  50. HAGB: Impurity effects • Impurities known to affect g.b. mobility strongly, depending on segregation and mobility. • CSL structures with good atomic fit less affected by solutes • Example: Pb bicrystals special general Rutter, J. W. and K. T. Aust (1960). “Kinetics of grain boundary migration in high-purity lead containing very small additions of silver and of gold.” Transactions of the Metallurgical Society of AIME218: 682-688. G.B. properties

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