1. LX2 Grain Boundary Properties: Mobility
27-750, Spring 2008
A.D. Rollett

2. References
Interfaces in Crystalline Materials, Sutton & Balluffi, Oxford U.P., Very complete compendium on interfaces.
Interfaces in Materials, J. Howe, Wiley, Useful general text at the upper undergraduate/graduate level.
Grain Boundary Migration in Metals, G. Gottstein and L. Shvindlerman, CRC Press, The most complete review on grain boundary migration and mobility.
Materials Interfaces: Atomic-Level Structure & Properties, D. Wolf & S. Yip, Chapman & Hall, 1992.

3. Outline Motivation, examples of anisotropic grain boundary properties
Grain boundary mobility
Basic theory
Experimental evidence

4. Grain Boundary Mobility
The mobility of grain boundaries is dominated by solute. Solutes tend to segregate to any interface and lower the free energy of the system (because of size misfit, for example). Therefore for a boundary to move away from the segregated solute requires energy to be supplied.
Mobility is also strongly sensitive to boundary type (i.e. atomic structure). High mobilities tend to be associated with a few CSL boundary types, e.g. 7 in fcc metals.
Mechanistic explanation is still lacking (and controversial!). Even a detailed understanding of mobility in pure systems is not available.

5. Mobility, contd. <111> Tilts
Highest mobility observed for <111> tilt boundaries. At low temperatures, the peaks occur at a few CSL types - S7, especially.
This behavior is inverse to that deduced from classical theory (Turnbull, Gleiter).
For stored energy driving force, strong tilt-twist anisotropy observed.
No simple theory available.
Atomistic simulation (MD) is in good agreement with experimental data.
<111> Tilts
general boundaries

6. G.B. Properties Overview: Mobility
If mobility depends on easy atomic transfer, does mobility scale with energy - MGB  (gGB )n? NO!
Consider the energies of boundaries in fcc metals based on the {111}-plane model for various low index axes. Note how the <100> twists have the highest energy, whereas experimentally, <111> tilts exhibit the highest mobility under all conditions.
Similar picture in bcc metals.

7. Mobility: LAGB characteristics
Low angle boundaries with dislocation structures are, in general, much less mobile than high boundaries. In broad terms, the diffusion distances are much larger in the case of LAGBs (diffusion between dislocations).
With only a few high mobility boundary types, the general picture is of very low mobility for LAGBs, moderate mobility for general boundaries, and the occasional peak for high angle boundaries.
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.

8. LAGB to HAGB Transitions
Read-Shockley for energy of low angle boundaries
Exponential function for transition in mobility from low- to high- angle boundaries

9. LAGB to HAGB Mobility Transitions
Transfer of atoms from the shrinking grain to the growing grain by atomic bulk diffusion mechanism
Low Angle Boundaries
Diffusion of vacancies (generally through the bulk) permits non-conservative motion (climb) of the dislocations in the grain boundary

10. 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
- Read, W. T. (1953). Dislocations in Crystals. New York, McGraw-Hill.

11. 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 grain growth, recrystallization etc., it gives a linear relation between force and velocity (as typically assumed).
Burke, J. E. and D. Turnbull (1952). Progress in Metal Physics 3: 220.

12. Burke-Turnbull
Given a difference in free energy 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,

13. 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

14. Linearity of migration rate against driving force
At what point is the linear relation no longer reasonable?
The criterion is the ratio of the driving force to the thermal activation energy, kT.
For a radius of curvature of 10nm, for example, and a g.b. energy of 1 J.m-2, ∆P=108 Pa. In terms of energy per atom in, say, Al, we multiply the volumetric pressure by the volume per atom, ~ 16 Å3, to obtain ∆P= J. Compare with kT ~ at 600°C. Clearly for very small scale microstructures (and low temperatures) we may expect the linearity to break down.

15. 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 Srolovitz group: [M. Upmanyu, D. Srolovitz and R. Smith, Int. Sci., 6, (1998) 41.
1. Zhang H, Mendelev MI, Srolovitz DJ Computer simulation of the elastically driven migration of a flat grain boundary. Acta mater. 52:
2. Zhang H, Mendelev MI, Srolovitz DJ Mobility of Sigma 5 tilt grain boundaries: Inclination dependence. Scripta mater. 52:
3. Zhang H, Srolovitz DJ Simulation and analysis of the migration mechanism of Sigma 5 tilt grain boundaries in an fcc metal. Acta mater. 54:
4. Zhang H, Srolovitz DJ, Douglas JF, Warren JA Characterization of atomic motion governing grain boundary migration. Phys. Rev. B 74.
5. Zhang H, Srolovitz DJ, Douglas JF, Warren JA Atomic motion during the migration of general [001] tilt grain boundaries in Ni. Acta mater. 55:
6. Zhang H, Upmanyu N, Srolovitz DJ Curvature driven grain boundary migration in aluminum: molecular dynamics simulations. Acta mater. 53:79-86.].
Much room for research!

16. HAGB Mobility: experiments
Following slides are taken from thesis research by Mitra Taheri.
Single crystals of aluminum (with alloy additions) are rolled to moderate reductions, a scratch is applied to induce nucleation of new grains along a line, and annealed to allow new grains to grow.
The growth rate of the recrystallizing grains, and the crystallography of their boundaries with the deformed matrix are studied.

17. Grain Morphology: HPAl+Zr at 350ºC
IPF images above show the annealing sequence from 187 to 237 minutes at 350°C
The most mobile grains (with ~38°<111>) exhibit faceting and are elongated
In the lower temperature annealing sequence of HPAl+Zr at 350C strong faceting was observed for many of the recrystallized boundaries, and were the most mobile boundares. It is shown (animation) by the pole figure analysis of a faceted edge that there is a 111 plane common to both the matrix and the growing grain. The pole figure also suggests that the leading edge is of pure tilt character whilst the sides are pure twist; this is consistent with the elongation observed and supports previous literature stating high mobility of tilt boundaries.
<111> pole figure suggests that the side facets of the highly mobile 38°<111> grain are sessile pure twist boundaries (111 planes)
Thesis research by M.Taheri

18. Grain Morphology: HPAl+Zr at 485ºC
In the higher temperature anneal of HPAl +Zr at 485C, grains don’t possess the faceting seen at 350C, which suggests a faceting-defaceting transition between 350C and 485C for this alloy.
IPF images at 8 and 20 minutes, respectively
Faceting not exhibited in the high temperature anneal of HPAl+Zr

19. Grain Boundary Mobility: HPAl+Zr
Looking at the boundary mobility as a function of misorientation angle (not specific to any boundary type), there is a transition from a sharp mobility maximum at 38 deg for the 350C anneal to a minimum at 38 w/2 maxima at 35 and 48deg for the 485C anneal. The transition in misorientation angle dependence of mobility suggests a transition in mobility boundary types. The 38 degree maxima in the 350C anneal corresponds to the 111 faceted boundaries shown in the previous micrographs.
Minimum Mobility: m4J-1s-1
Minimum Mobility: m4J-1s-1
Change from low to high annealing temperature yields peak shift: maximum at 38º shifts to minimum with 2 maxima at 35º and 48º

20. Mobility vs. Boundary Type: 350ºC
R3 = 0 Projection R2 R1
<111>, <110>
[Randle]
The dependence of mobility on boundary type was plotted in Rodrigues space. To the left is a schematic diagram of where the boundary types and CSL boundaries are located in each section of the plot on an R3=0 section. This section corresponds to the bottom left section of the plotted data. It is seen that there are practically no <001> boundaries. Looking to the other sections, <111> boundaries can be found on the bottom left corner of each colored section in each triangle while <110> boundaries occur along the hypoteneuse of the R3=0 section. Looking at the top left section of the plotted data, it is evident that there is a strong sigma 7 mobility maxima. R1
<100>
At 350ºC, only boundaries close to 38°<111> are mobile

21. Mobility in Rodrigues space: 350 vs. 485ºC
When moving to a higher annealing temperature, <001> boundaries exhibit a higher mobility, as shown in the bottom left section of the plotted data, where a mobility peak for sigma 5 boundaries is observed. The presence of mobility peaks for <111> boundary types is still evident in the R3=.149 section (top left corner) and in the R3=.099 section just under that. These plots support the shift in misorientation angle dependent mobility maxima and the faceting- defaceting transition for the temperature change. S7 S7
At 485ºC, other boundary types become mobile: <100>, others, and the peak near 38°<111> splits.

22. Commercial Purity Al + Zr
At 485C, presence of mobility peaks for <111> and <110> boundary types is evident. The presence of <110> boundaries along the hypoteneuse of the R3=0 section is evident while not in the HP+Zr. Specifically, the preference seems to be at sigma 7, 49, and 11. S7 S7
Broad range of mobile boundary types, with peaks near <111>.
Some near-<100> mobility appears, with minor 38°<111> peak.

23. Compensation Effect 40° or 36° <111> 38°<111>
Both curvature and stored energy driving forces appear to yield similar results. The apparent activation enthalpy varies significantly, leading to a “compensation effect.”
Huang, Humphreys et al.
Gottstein, Shvindlerman et al.

24. Compensation Effect
The compensation effect can be understood in terms of a proportionality between the enthalpy (∆H) and entropy (∆S) associated with a process, in this case, grain boundary migration.

25. Activation Energy for GB Migration in Al
A minimum in activation energy near 38º<111> is apparent for both the present work and from the literature.
Note: Gottstein et al. and Molodov et al. studied Al bicrystals under curvature driving force. “This work” refers to experiments by Taheri with stored energy as the driving force. “Simulation” refers to molecular dynamics simulations by Zhang with an interatomic potential to represent Al.
curvature
Stored E
simulation

26. Solute effect on HAGB Mobility
Solutes play a major role in g.b. mobility by reducing absolute mobilities, even at very low levels
Simulations typically have no impurities included: therefore they model ultra-pure material
Simulations of mobility typically show (much) lower activation energies than those measure experimentally

27. 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 [Aust]
special
general

28. HAGB Mobility: impurity effect on recrystallization kinetics
increasing Cu content
R. Vandermeer and P. Gordon, Proc. Symposium on the Recovery and Recrystallization of Metals, New York, TMS AIME, (1962) p. 211.

29. HAGB Mobility: impurity effect on recrystallization kinetics
decreasing Fe content = increasing mobility
V (cm.s-1)
1/T
F. R. Boutin, J. Physique, C4, (1975) C4.355.

30. Impurity (solute) effect on mobility, contd.
Example of Ga additions to Al (LHS) show that at low levels, certain solutes can increase mobility. Adding 10ppm Ga to % Al increases the mobility whereas adding 410ppm Ga to Al decreases the mobility (as expected).
Note the low levels of solute that have measurable effects on mobility. Example (RHS) of adding Cu to Al shows an effect at a/o, i.e. at the ppm level.
Main points:
Solutes affect grain boundary migration at v low levels.
Solutes can sometimes appear to increase mobility, not just decrease it as theory predicts.
Gottstein & Shvindlerman: grain boundary mobility in Al.
Gordon & Vandermeer: impurities in aluminum

31. Impact of Mobility on Evolution
What impact does the anisotropy of mobility have on microstructural evolution?
None on interface texture (GBCD), to 1st order, if the texture is random (based on moving finite element simulations, PhD thesis by Jason Gruber)
Strong effect on texture development (grain texture) is there is a non-random texture present to begin with.
Strongest effect known for grain growth
Appreciable but less important effect in recrystallization (PhD thesis by Abhijit Brahme)

32. 3D Grain Growth Simulation (MC)
Strong fcc rolling texture (FC, e = 2) with ~6% cube added
Copper component: {112}<111> ND TD
3cub13; E&M; T=0.5; Emin=0.55 RD

33. Grain Boundary Energy, Mobility
Energy: only the Read-Shockley equation for grain boundary energy at small misorientations was used.
Mobility: only the peak mobility at 40°<111> was used.
m= also a peak w.r.t. axis

34. Texture Components Dillamore 2° spread; 6% initial cube fraction.
Copper: monotonic decrease
S: initial increase before decrease
Goss: marked increase before cube dominates
Brass: first to decrease
Cube: monotonic increase
copper {112}<111>
cube {001}<100>
S {412}<634>
Dillamore
Goss {011}<100>
brass {110}<112>
plot '3cub32.comps' using 1:2 w l lw 3, '3cub32.comps' using 1:3 w l lw 3, '3cub32.comps' using 1:4 w l lw 3, '3cub32.comps' using 1:5 w l lw 3, '3cub32.comps' using 1:6 w l lw 3, '3cub32.comps' using 1:7 w l lw 3

35. Peak:Plateau Ratio
The probability of the cube component becoming dominant increases with increasing peak:plateau ratio (smaller number). This corresponds roughly to either increasing temperature or to increasing solute levels.
Average growth rate decreases because the average mobility in the system decreases with increasing ratio.

36. Mechanical Stress as Driving Force on Grain Boundaries
The work of Dr. Myrjam Winning at the RWTH Aachen has shown, remarkably enough, that mechanical stresses can cause grain boundaries to migrate even for high angle grain boundaries, for which no dislocation structure has been observed.
The following slides were provided by Dr. Winning to illustrate the effect.

37. Mechanical stresses and grain boundaries
Systematic investigations of interactions between
grain boundaries and mechanical stress fields.
individual and planar grain boundaries
no interactions of other grain boundaries
no other driving forces
constant grain boundary structure
bicrystals (high purity Al) with planar grain boundaries +
constant shear stress
in-situ observation of grain boundary motion

38. Determination of gb position by X-ray diffraction
Grain I
in Bragg-position
Grain II
not in
Bragg-position
Area of
X-ray spot
(350µm x 800µm)
Grain boundary

39. Determination of gb position by X-ray diffractionIm Iu x

40. Determination of gb position by X-ray diffractionIm Iu x

41. Determination of gb position by X-ray diffractionIm Iu x

42. Determination of gb position by X-ray diffractionIm Iu x

43. Determination of gb position by X-ray diffractionIm Iu x

44. Determination of gb position by X-ray diffractionIm Iu x

45. Determination of gb position by X-ray diffractionIm Iu x

46. Determination of gb position by X-ray diffractionIm Iu x

47. Determination of gb position by X-ray diffractionIm Iu x

48. Determination of gb position by X-ray diffractionIm Iu x

49. Determination of gb position by X-ray diffractionIm Iu x

50. Determination of gb position by X-ray diffractionIm Iu x

51. Determination of gb position by X-ray diffractionIm Iu x

52. Determination of gb position by X-ray diffractionIm Iu x

53. Determination of gb position by X-ray diffractionIm Iu x

54. Determination of gb position by X-ray diffractionIm Iu x

55. Determination of gb position by X-ray diffraction
Migration of grain boundary
requires movement of the
sample => speed of sample
is the same as velocity
of grain boundary I0 Im Iu x

56. XICTD*
detector
sample
X-ray source
Dr. Molodov already showed you in his talk the principle of the in-situ measurement of the grain boundary motion and the main components of the x-ray interface continuous tracking device.
In this picture you can see the x-ray tube on the rigjht side which is fixed.
On the left side the detector. The detector can be rotated around the specimen holder in the middle.
In this direction we follow the grain boundary motion by the movement of the specimen.
The specimen motion can be controlled by a step motor here on the right side.
At last you can see that the specimen chamber is closed to exposed the sample to a nitrogen atmosphere to avoid thermal grooving on the boundary.
*X-ray interface continuous tracking device

57. XICTD
spring
sample
thermo couple
grain boundary

58. Stress-induced grain boundary motion
<112>-tilt axis
q=12.9°
time [s]
grain boundary position [µm]
M. Winning, G. Gottstein, L.S. Shvindlerman
Acta Materialia ,

59. Macroscopic evidence for grain boundary motion
<111>-tilt grain boundary
with q=16.0°
400µm
390µm
150µm
300µm
<100>-tilt grain boundary
with q=10.6°
Swept distance: 390µm
Measured distance (in-situ): 365µm
Swept distance: 300µm
Measured distance (in-situ): 293µm
M. Winning, G. Gottstein, L.S. Shvindlerman
Acta Materialia ,
M. Winning,
Acta Materialia ,

60. Arrhenius diagram LAGBs
Temperature [K]
temperature [K]
625
900
825
725
<111>
DH = 1.29eV
lnm0 = 23.99
<112>
DH = 1.28eV
lnm0 = 23.07
<100>
DH = 1.19eV
lnm0 = 19.32
reciprocal temperature [1/K]

61. Arrhenius diagram HAGBs
700
750
800
850
temperature [K]
<112>
DH = 0.81eV
lnm0 = 16.47
<111>
DH = 0.74eV
lnm0 = 14.26
<100>
lnm0 = 13.63
Temperature [K]
reciprocal temperature [1/K]

62. Activation parameters
<112>
<111>
<100>
13.6°
8.6° 5 10 15 20 25 30 35
0.5
0.6
0.7
0.8
0.9 1
1.1
1.2
1.3
1.4
1.5 49 59 69 79 89 99
109
119
129
139
misorientation angle [°]
activation enthalpy [eV]
activation enthalpy [kJ/mol]
M. Winning,
Acta Materialia ,
M. Winning, G. Gottstein, L.S. Shvindlerman
Acta Materialia ,

63. Review of Bicrystal Experiments
Results of bicrystal experiments
Review of Bicrystal Experiments
Mechanical stress activates motion of planar grain boundaries.
Activation enthalpy for tilt grain boundaries ~ eV (LAGB) and
~ eV (HAGB).
Migration mechanism of planar grain boundaries can be explained
by motion of dislocations in the grain boundaries.
Sharp transition from LAGB to HAGB:
qtrans=13.6° for <112>/<111> tilt grain boundaries
qtrans=8.6° for <100> tilt grain boundaries.
M. Winning, G. Gottstein, L.S. Shvindlerman
Acta Materialia ,
M. Winning, G. Gottstein, L.S. Shvindlerman
Acta Materialia ,
M. Winning,
Acta Materialia ,
M. Winning, A.D. Rollett
Acta Materialia ,

64. 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.
In fcc metals, the most mobile boundaries (at low temperatures) are near to 38°<111>. No theory explaining this maximum (or its variations) exists - a challenge for research!
Inversions of ranking in mobility can occur with changes in temperature.

65. Summary, contd.
Anisotropy of mobility is similar regardless of driving force.
Mobilities vary over orders of magnitude, in contrast to energies that only vary by about a factor of 2.
Solutes strongly decrease boundary mobilities, even at low concentrations.
Mobility is much more complex than energy and little exists beyond the basic activated rate theory of Burke & Turnbull. Computer simulation shows similar trends to experimental observations.
Mechanical driving force also causes high angle boundaries to migrate, despite the lack of an observable dislocation structure. This behavior also seen in MD simulations by Cahn, Mishin.