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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

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

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

LAGB to HAGB Transition

- LAGB: steep risewith angle.HAGB: plateau

Disordered Structure

Dislocation Structure

G.B. properties

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

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), whereE0 = µb/4π(1-n); A0 = 1 + ln(b/2πr0)

G.B. properties

1.1 LAGB experimental results

- Experimental results on copper.

[Gjostein & Rhines, Acta metall. 7, 319 (1959)]

G.B. properties

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

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.

[001]

0.33

0.30

0.26

0.23

[111]

[101]

Low Angle Grain Boundary Energy

High

[117]

[105]

[113]

[205]

[215]

[335]

[203]

Low

[8411]

[323]

[727]

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

vs.

G.B. properties

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

1.2 Exptl. Observations

<100>Tilts

Twin

<110>Tilts

G.B. properties

Hasson, G. C. and C. Goux (1971). “Interfacial energies of tilt boundaries in aluminum. Experimental and theoretical determination.” Scripta metallurgica5: 889-894

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.

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

Atomistic models: results

- Results of atomistic modeling confirm the importance of the more symmetric boundaries.

G.B. properties

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

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

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

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

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

h

dx

Tilt Boundary MotionBurgers 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

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.

G.B. properties

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.

G.B. properties

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

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

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

Dislocation Modelsfor Low Angle G.B.s

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

G.B. properties

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.

G.B. properties

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.

G.B. properties

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

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

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.

G.B. properties

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

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

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

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

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

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

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 (˚)

G.B. properties

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

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)

G.B. properties

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)

G.B. properties

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.

G.B. properties

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

Log10M

(µm4/s MPa)

Log10M

(µm4/s MPa)

4

32

1

q (˚)

G.B. properties

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

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

LAGB to HAGB Transitions

- Read-Shockley forenergy of low angleboundaries
- Exponentialfunction for transitionfrom low- to high-angle boundaries

G.B. properties

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

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 AIME218: 682-688.

G.B. properties

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 Physics3: 220.

graduate

G.B. properties

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

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

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

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,...

G.B. properties

HAGB M: Boundary velocity

Simulation Experiment

Steady-state migration + initial and final transients

G.B. properties

S7

S13

Mobility M

Grain Boundary Energy g

Misorientation q

Misorientation q

HAGB M: simulation results- Extract boundary energy from total energy vs. half-loop height(assume constant entropy)
- M=M*/g

G.B. properties

HAGB M: Activation energy

simulation

S19

S7

S13

specialboundary

Q (e)

experiment

S7

Lattice diffusion between dislocations

Q (eV)

G.B. properties

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

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

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

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