1

Supplemental SlidesSupplemental Slides

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

2

Experimental Impact of EnergyExperimental 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 comparisonWetting 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. propertiesLow-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

vs.

Δg∧

Low Angle Grain Boundary Energy, Yang et al.

High

Low

[335]

[323][727]

[203]

[205]

[105]

[215]

[117]

[113]

[8411]

[001]

[101][111]

0.30

0.26

0.23

0.33

"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 Materiala 44: 2735-2740 (2001).

2 4 6 8 10 12 14 16 180.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Relative Energy

Misorientation Angle

6

Dislocation models of HAGBsDislocation models of HAGBs• Boundaries near CSL points expected to

exhibit dislocation networks, which is observed.

<100> twists

7

Atomistic modelingAtomistic 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: resultsAtomistic models: results

• 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.

g.b. plane

Wolf & Yip

9

Coordination NumberCoordination NumberReasonable 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 MobilityLow 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*=kc, 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

M vs. Δg∧

[001]

[101]

[111]

[117]

[113]

[335]

[105]

[205]

[203]

[215]

[8411]

[727] [323]

0.30.1

0.0004

0.9

Relative Mobility

0.030.01

LAGB Mobility in Al, experimental

High

Low

12

LAGB: Axis DependenceLAGB: 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 <111> 12.4o grain boundary=> dislocations are nearly parallel to the boundary normal

=> = /2

Symmetrical <001> 11.4o grain boundary=> nearly 45o

alignment of dislocations with respect to the boundary normal

=> = 45o +/2

14

2.1 Low Angle GB Mobility, contd.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 Dislocation ModelsModels

for Low Angle for Low Angle G.B.sG.B.s

Sutton and Balluffi (1995). Interfaces in Crystalline Materials. Clarendon Press, Oxford, UK.

16

Theory: DiffusionTheory: Diffusion• Atom flux, J, between the dislocations is:

where DL is the atom diffusivity (vacancy mechanism) in the lattice; is the chemical potential;kT is the thermal energy;and is an atomic volume.

J =−DL

ΩkT∇μ

17

Driving ForceDriving Force• A stress 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.

∇μ ≅2τΩd

18

Atom FluxAtom 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].

I =2AJ ≅2DLτkT

19

Dislocation VelocityDislocation 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.

v=I2Ωb

≅2 2DLΩkTb

τ

20

Mobility (Lattice Diffusion Mobility (Lattice Diffusion only)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/√2). • 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.

p=τ b/ 2( ) d=τθ

M=2 2DLΩkTbθ

21

Theory: Addition of a Pipe Diffusion Theory: Addition of a Pipe Diffusion ModelModel

• 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:

J ⊥ =−D⊥

ΩkT∇μ

22

Lattice+Pipe DiffusionLattice+Pipe Diffusion

• The total current of atoms from one dislocation parallel to the tilt axis to the next (per length of boundary) is

where 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.

I =2AJ +J ⊥

πδ2

d2

≅2τkT

DL +πD⊥δ

2

d1d2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

23

Boundary VelocityBoundary Velocity

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

v=I2Ωb

≅2 2ΩkTb

τ DL +πD⊥δ

2

d1d2

⎛

⎝ ⎜

⎞

⎠ ⎟

≅4ΩkTb

DL

2+πD⊥δ

2θbd2

⎛

⎝ ⎜

⎞

⎠ ⎟ τ

24

Mobility (Lattice Mobility (Lattice andand Pipe Diffusion) Pipe Diffusion)

• The mobility M=v/() is now simply:

This expression suggests that the mobility increases as the spacing between dislocations perpendicular to the tilt axis decreases.

M≅4ΩkTb

DL

2θ+πD⊥δ

2

bd2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

25

Effect of twist angleEffect of twist angle

• If the density of dislocations running perpendicular to the tilt axis is associated with a twist component, then:

where 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.

M≅4ΩkTb

DL

2θ+πD⊥δ

2

b2 φ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

26

Effect of MisorientationEffect of Misorientation

• If the density of the perpendicular dislocations is proportional to the density of parallel ones, then the mobility is:

where is a proportionality factor. Note the combination of mobility increasing and decreasing with misorientation.

M ≅4ΩkTb

DL

2 fLθ+α

πD⊥δ2

f⊥b2 θ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

27

Results: Ni MobilityResults: Ni Mobility

• Nickel: QL=2.86 eV, Q=0.6QL, D0L=D0=10-4 m2/s, b=3x10-10 m, =b3, =b, =1, k=8.6171x10-5 eV/K.

T (˚K)

(˚)

M(1

0-10

m4 /

[J s

])

28

Theory: Reduced MobilityTheory: Reduced Mobility

• Product of the two quantities M*=M 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:

M* =Mγ ≅−4ΩkTb

DL

2+α

πD⊥δ2

b2 θ2⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ln

θθ*

⎛ ⎝ ⎜

⎞ ⎠ ⎟

29

Results: Ni Reduced MobilityResults: Ni Reduced Mobility

• 0=1 J/m2 and *=25˚, corresponding to a maximum in the boundary mobility at 9.2˚.

log

10M

* (1

0-11m

2/s

)

(˚) T (˚K)

30

Results: AluminumResults: AluminumMobility vs. T and Mobility vs. T and

The vertical axis is Log10 M.

0 = 324 mJ/m2, *= 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, b, b = 0.286 nm, mb3/√2, = 1.

log

10M

(µm

4/s

MP

a)

(˚) T (K)

31

Comparison with Expt.:Comparison with Expt.: Mobility vs. Angle at 873K Mobility vs. Angle at 873K

M. Winning, G. Gottstein & L.S. Shvindlerman, Grain Boundary Dynamics under the Influence of MechanicalStresses, Risø-21 “Recrystallization”, p.645, 2000.

(˚)

Lo

g10

M(µ

m4 /

s M

Pa)

0

-1

-2

-3

-4

-5

Lo

g10

M(µ

m4 /

s M

Pa)

32

Comparison with Expt.: Comparison with Expt.: Mobility vs. Angle at 473K Mobility vs. Angle at 473K

(˚)

Lo

g10

M(µ

m4 /

s M

Pa)

4

3

2

1

Lo

g10

M(µ

m4 /

s M

Pa)

33

Discussion on LAGB mobilityDiscussion 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; conclusion2.1 LAGB mobility; conclusion

• Agreement between calculated (reduced) mobility and experimental results is remarkably good. Only one (structure sensitive) adjustable parameter (), 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

HAGB Mobility: the U-bicrystalHAGB 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

vV

w

Dunn, Shvindlerman, Gottstein,...

36

HAGB M: Boundary velocityHAGB M: Boundary velocity

Steady-state migration + initial and final transients

Simulation Experiment

37

HAGB M: 2D simulation resultsHAGB M: 2D simulation resultsG

rain

Bo

un

dar

y E

ner

gy

Misorientation Misorientation

Mo

bili

ty M

• Extract boundary energy from total energy vs. half-

loop height (assume constant entropy)

• M=M*/

Note: misorientation angle shown in plots is 1/2 of total angle.

38

HAGB M: HAGB M: Activation Activation

energyenergy

M* =Mo*e−Q/kT

simulation

experiment

Lattice diffusion between dislocations

specialboundary

Q

()

Q (

eV)

Simulations exhibit much smaller activation energies than experiments, possibly because solutes affect experimental results.

39

3D simulations: reduced mobility (M*) vs. Misorientation

13 7(m

4/J

s)

(deg)

Zhang, Upmanyu, Srolovitz

40

13 7

Mobility and Energy vs. Misorientation

713

(J/m

2)

(m4

/Js

)(deg) (deg)

Zhang, Upmanyu, Srolovitz

41

Mobility vs. Misorientation13 7

(m4/Js)

(deg)

Zhang, Upmanyu, Srolovitz

42

Reduced Mobility, M*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 MobilityHigh 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 7 CSL type).

• Note broad maximum.

Gottstein & Shvindlerman: grain boundary migration in metals

44

Mobility of HAGBs with stored Mobility of HAGBs with stored energy driving forceenergy 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

Simulation Results: Misorientation Axis Dependence

<111> tilt misorientations = fastest moving boundariesDramatic decrease in mobility with deviation in tilt axis

T = 0.7Tm

001 (45o)

110 (54.7o)

112 (56.9o)

023 (50.7o)

111 (54.7o)

144 (30.0o) 122 (55.3o)

Unpublished work by Upmanyu