G.B. properties1 Grain Boundary Properties: Energy, Mobility 27-765, Spring 2001 A.D. Rollett.

G.B. properties 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

Dislocation Structure

Disordered 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(/2)/b /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):

gb = E0 ln, whereµb/4π(1-); 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):

gb = sin {Ucore/b - µb2/4π(1-)ln(sin}

• Note: this form of energy variation may also be applied to CSL-vicinal boundaries.

G.B. properties 11 vs. Δg∧

[001]

[101][111]

0.30

0.26

0.23

0.33

Low Angle Grain Boundary Energy

A. Otsuki, Ph.D.thesis, Kyoto University, Japan (1990)

Energy (mJ/m2),T=240oC [001] [101] [111]

Tilt 190 170 148

Twist 200 205 155

Misorientation Axis [uvw] ; =5

o

High

Low

[335]

[323][727]

[203]

[205]

[105]

[215]

[117]

[113]

[8411]

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.

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

Hasson, G. C. and C. Goux (1971). “Interfacial energies of tilt boundaries in aluminum. Experimental and theoretical determination.” Scripta metallurgica 5: 889-894

<100>Tilts

<110>Tilts

Twin

G.B. properties 14

Dislocation models of HAGBs• Boundaries near CSL points expected to

exhibit dislocation networks, which is observed.

<100> twistsHowe, J. M. (1997). Interfaces in Materials. New York, Wiley Interscience.

G.B. properties 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 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.

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30

Misorientation Angle (degrees)

Relative Boundary Mobility

High Angle Boundaries

Transfer of vacancies between two adjacent sets of dislocations by grain boundary diffusion mechanism

Low Angle Boundaries

Transfer of atoms from the shrinking grain to the growing grain by atomic bulk diffusion mechanism

LA->HAGB Transition

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

(Bauer and Lanxner, Proc. JIMIS-4 (1986) 411)

Burgers vectors inclined with respect to the boundary plane in proportion to the misorientation angle.

glide climb

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*=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.

G.B. properties 24M 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

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 <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

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; is the chemical potential;kT is the thermal energy;and is an atomic volume.

J =−DL

ΩkT∇μ

G.B. properties 30

Driving 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

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].

I =2AJ ≅2DLτkT

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.

v =I2Ωb

≅2 2DLΩ

kTbτ

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/√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 2DLΩ

kTbθ

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: J ⊥ =−

D⊥

ΩkT∇μ

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 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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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:

v=I2Ωb

≅2 2ΩkTb

τ DL +πD⊥δ2

d1d2

⎛

⎝ ⎜

⎞

⎠ ⎟

≅4ΩkTb

DL

2+

πD⊥δ2θbd2

⎛

⎝ ⎜

⎞

⎠ ⎟ τ

G.B. properties 37

Mobility (Lattice and 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⊥δ2

bd2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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 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⊥δ2

b2 φ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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 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

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

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, =b3, =b, =1, k=8.6171x10-5 eV/K.

T (˚K)

(˚)

M(1

0-10 m

4 /[J

s]

)

G.B. properties 41

Theory: 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*

⎛ ⎝ ⎜

⎞ ⎠ ⎟

G.B. properties 42

Results: Ni Reduced Mobility

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

log 10

M*

(10-1

1 m2 /

s)

(˚) T (˚K)

G.B. properties 43

Results: AluminumMobility 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, 5mb3/√2, = 1.

log 10

M(µ

m4 /

s M

Pa)

(˚) T (K)

G.B. properties 44

Comparison with Expt.: 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.

(˚)

Log

10M

(µm

4 /s

MP

a)

0

-1

-2

-3

-4

-5

Log

10M

(µm

4 /s

MP

a)

G.B. properties 45

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

(˚)

Log

10M

(µm

4 /s

MP

a)

4

3

2

1

Log

10M

(µm

4 /s

MP

a)

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 (), 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

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

EnergyMobility

Angle (°)

c2=1.-0.99*exp(-.5*(c0/15)^9)

c1=c0/15.*(1.-ln(c0/15.))• 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 7 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

generalRutter, 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: 682-688.

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.

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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:

v=b ν+exp −ΔGm

+

kT

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪ −ν−exp −

ΔGm−+ΔPb3

kT

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Given similar attack frequencies and activation energies in both directions,

graduate

G.B. properties 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.

v=bνDebyeexp−ΔGmkT

⎧ ⎨ ⎩

⎫ ⎬ ⎭

1−exp −ΔPb3

kT

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

v=b4νDkT

exp −ΔGmkT

⎧ ⎨ ⎩

⎫ ⎬ ⎭ ΔP

Mobility

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G.B. properties 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!

graduate

G.B. properties 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

vV

w

Dunn, Shvindlerman, Gottstein,...

G.B. properties 56

HAGB M: Boundary velocity

Steady-state migration + initial and final transients

Simulation Experiment

G.B. properties 57

HAGB M: simulation resultsG

rain

Bou

nd

ary

En

ergy

Misorientation Misorientation

Mob

ilit

y M

• Extract boundary energy from total energy vs.

half-loop height (assume constant entropy)

• M=M*/

G.B. properties 58

HAGB M: Activation

energy

M* =Mo*e−Q/kT

simulation

experiment

Lattice diffusion between dislocations

specialboundary

Q

()

Q (

eV)

G.B. properties 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

R. Vandermeer and P. Gordon, Proc. Symposium on the Recovery and Recrystallization of Metals, New York, TMS AIME, (1962) p. 211.

F. R. Boutin, J. Physique, C4, (1975) C4.355.

V (cm.s-1)

1/T

decreasing Fe content

increasing Cu content

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.