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 rise with 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. 1.1 Tilt boundary b 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), where E0 = 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)]
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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. Low Angle Grain Boundary Energy
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.
Low Angle Grain Boundary Energy
[001]
[101]
[111]
High
[117]
[105]
0.30
0.26
0.23
0.33
[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 <110> Tilts Twin
G.B. properties
Hasson, G. C. and C. Goux (1971). “Interfacial energies of tilt boundaries in aluminum. Experimental and theoretical determination.” Scripta metallurgica 5:

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. Tilt Boundary Motion h 
boundary displacement dx
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= 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. LAGB Mobility in Al, experimental
[001]
0.3
0.1
0.0004
0.9
Relative
Mobility
0.03
0.01
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 Models for 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 W is 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 d is 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,
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 (˚)
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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)
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43. Results: Aluminum Mobility vs. T and q
The vertical axis is Log10 M.
log10M
(µm4/s MPa)
g0 = 324 mJ/m2, q*= 15°, DL(T) exp-{ J/mol/RT} m2/s, D(T) exp-{81855 J/mol/RT} m2/s, d=b, b = nm, W = m3 = b3/√2, a = 1.
q (˚)
T (K)
G.B. properties

44. Comparison with Expt.: Mobility vs. Angle at 873K
Log10M
(µm4/s MPa)
-1 -2 -3 -4 -5
Log10M
(µm4/s MPa)
q (˚)
M. Winning, G. Gottstein & L.S. Shvindlerman, Grain Boundary Dynamics under the Influence of Mechanical Stresses, 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
3 2 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 for energy of low angle boundaries
Exponential function for transition from 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 AIME 218:
G.B. properties

51. HAGB mobility: theory
The standard theory for HAGB mobility is due to Burke & Turnbull, based on thermally activated atomic transfer across the interface.
For the low driving forces typical in grian growth, recrystallization etc., it gives a linear relation between force and velocity (as typically assumed).
Burke, J. and D. Turnbull (1952). Progress in Metal Physics 3: 220.
graduate
G.B. properties

52. Burke-Turnbull
Given a difference in free energy (per unit volume) for an atom attached to one side of the boundary versus the other, ∆P, the rate at which the boundary moves is:
Given similar attack frequencies and activation energies in both directions,
G.B. properties
graduate

53. Velocity Linear in Driving Force
Then, for small driving forces compared to the activation energy for migration, ∆Pb3«kT, which allows us to linearize the exponential term.
Mobility
G.B. properties
graduate

54. HAGB Mobility
The basic Burke-Turnbull theory ignores details of g.b. structure:
The terrace-ledge-kink model may be useful; the density of sites for detachment and attachment of atoms can modify the pre-factor.
Atomistic modeling is starting to play a role: see work by Upmanyu & Srolovitz [M. Upmanyu, D. Srolovitz and R. Smith, Int. Sci., 6, (1998) 41.].
Much room for research!
G.B. properties
graduate

55. 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 x y v V w
Dunn, Shvindlerman, Gottstein,...
G.B. properties

56. HAGB M: Boundary velocity
Simulation Experiment
Steady-state migration + initial and final transients
G.B. properties

57. HAGB M: simulation results
Extract boundary energy from total energy vs. half-loop height (assume constant entropy)
M=M*/g
Grain Boundary Energy g
Misorientation q
Misorientation q
Mobility M S7
S13
S19
G.B. properties

58. HAGB M: Activation energy
simulation
S19 S7
S13
special boundary
Q (e)
experiment S7
Lattice diffusion between dislocations
Q (eV)
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59. HAGB M: Issues; “dirt”
Solutes play a major role in g.b. mobility by reducing absolute mobilities at very low levels.
Simulations typically have no impurities included: therefore they model ultra-pure material.
G.B. properties

60. HAGB M: impurity effect on recrystallization
increasing Cu content
V (cm.s-1)
decreasing Fe content
1/T
F. R. Boutin, J. Physique, C4, (1975) C4.355.
R. Vandermeer and P. Gordon, Proc. Symposium on the Recovery and Recrystallization of Metals, New York, TMS AIME, (1962) p. 211.
G.B. properties

61. GB Mobility: Summary
The properties of low angle grain boundaries are dictated by their discrete dislocation structure: energy logarithmic with angle; mobility exponential with angle.
The kinetic properties of high angle boundaries are (approx.) plateau dictated by local atomic transfer. Special boundary types have low energy and high/low mobility.
G.B. properties