Supplemental Slides

1. Supplemental Slides • In the slides that follow, various details are given that it was not possible to cover in the lecture

2. 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 et al., ICOTOM-12 (1999).

3. Wetting comparison • High energy (light) boundaries should be wet (“W”). • Low energy boundaries (dark) should be dry (“U”). • Example of Cu wetting boundaries in Fe with (311) on one side. • Takashima, M., P. Wynblatt, and B.L. Adams, Correlation of grain boundary character with wetting behavior. Interface Science, 2000. 8: p. 351-361.]

4. Low-angle g.b. properties • Recently, the properties of low angle grain boundaries have been measured by the MIMP at CMU. • The results confirm the Read-Shockley relationship. • A variation of energy with misorientation axis was also found: boundaries with <111> misorientation axes had the lowest energies whereas those with <100> axes had the highest. The variation was only over a range of +/- 10%, however.

5. [001] 0.33 0.30 0.26 0.23 [111] [101] Low Angle Grain Boundary Energy, Yang et al. High [117] [105] [113] [205] [215] [335] [203] Low [8411] [323] [727] "Measuring relative grain boundary energies and mobilities in an aluminum foil from triple junction geometry", C.-C. Yang, W. W. Mullins and A. D. Rollett, Scripta Materiala44: 2735-2740 (2001).  vs.

6. Dislocation models of HAGBs • Boundaries near CSL points expected to exhibit dislocation networks, which is observed. <100> twists

7. 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. • Grain boundaries in fcc metals: Cu, Au

8. Atomistic models: results g.b. plane • Results of atomistic modeling confirm the importance of the more symmetric boundaries. • Example of symmetric tilt boundary energy for embedded-atom-method calculations using either Lennard-Jones, copper or gold interatomic potentials. Wolf & Yip

9. 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. Wolf & Yip

10. 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.

11. 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.

12. 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.

13. 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

14. 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°).

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

16. 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.

17. 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.

18. 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].

19. 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.

20. 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.

21. 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:

22. 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.

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

24. 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.

25. 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.

26. 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.

27. 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 (˚)

28. 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:

29. 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)

30. 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)

31. 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.

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

33. 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.

34. 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.

35. y v V w x HAGB Mobility: the U-bicrystal • The curvature of the end of the interior grain is constant (unless anisotropy causes a change in shape) and the curvature on the sides is zero. • Migration of the boundary does not change the driving force • Simulation and experiment Dunn, Shvindlerman, Gottstein,...

36. HAGB M: Boundary velocity Simulation Experiment Steady-state migration + initial and final transients

37. S19 S7 S13 Mobility M Grain Boundary Energy g Misorientation q Misorientation q HAGB M: 2D simulation results • Extract boundary energy from total energy vs. half-loop height(assume constant entropy) • M=M*/g Note: misorientation angle shown in plots is 1/2 of total angle.

38. simulation HAGB M: Activation energy S19 S7 S13 specialboundary Q (e) experiment S7 Lattice diffusion between dislocations Q (eV) Simulations exhibit much smaller activation energies than experiments, possibly because solutes affect experimental results.

39. S13 S7 3D simulations: reduced mobility (M*) vs. Misorientation (m4/Js) (deg) Zhang, Upmanyu, Srolovitz

40. S13 S13 S7 S7 Mobility and Energy vs. Misorientation (J/m2) (m4/Js) (deg) (deg) Zhang, Upmanyu, Srolovitz

41. S13 S7 Mobility vs. Misorientation (m4/Js) (deg) Zhang, Upmanyu, Srolovitz

42. Reduced Mobility, M* • In many experiments on g.b. mobility, only the migration rate can be measured and the boundary curvature. If the energy of the boundary is not known (or must be assumed to be constant) then one can only derive the reduced mobility, M*. Where M and E are the mobility and energy defined in the standard way, M* = M E.

43. 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

44. Mobility of HAGBs with stored energy driving force • Huang & Humphreys, The effect of solute elements on grain boundary mobility during recrystallization of single-phase aluminum alloys, Proc. Conf. Rex & Gr.Gr., Aachen, vol.1 409 (2001). • As previously observed, broad peak in mobility observed centered on 40° <111> misorientation with 10° FWHM w.r.t. misorientation angle. Similar decrease with deviation from <111> axis.

45. 111 (54.7o) 110 (54.7o) 023 (50.7o) 001 (45o) 112 (56.9o) 144 (30.0o) 122 (55.3o) Simulation Results: Misorientation Axis Dependence T = 0.7Tm <111> tilt misorientations = fastest moving boundaries Dramatic decrease in mobility with deviation in tilt axis Unpublished work by Upmanyu