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Interphase

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Low angle

Semicoherent

High angle

Incoherent

Based on angle of rotation

Twist

Interphase

Tilt

Based on axis

Mixed

Based on Lattice Models

Special

Epitaxial/Coherent

Random

Based on Geometryof the Boundary plane

Curved

Wulff-type constructions

Faceted

Mixed

- Frankel-Kontorova (Frank-Van der Merwe) model
- Localization of distortions (dislocations) in commensurate case
- Localization & Aubry Transition for incommensurate case

- Vernier
- Rotated Registries
- Co-incidence of Reciprocal Lattice Approach (Fletcher-Lodge; Near Coincident Site Model)

See additional reading in Dislocations/Grain Boundary directories for original papers

Interface Bonding

Elastic distortions

Elastic Distortions

E = S [W(xi-xi-1) + V(xi)]

V(x) = Interface Bonding

W(x) = Elastic Energy

F. Frank & J. H. van der Merwe, Proc. R. Soc. London Ser. A 198, 205 (1949) J. P. Hirth & J. Lothe, Theory of Dislocations, Krieger Publishing Company, Malabar, 1982 .

Y. Frenkel & T. Kontorova, Z. Exp. Th. Phys. 8, 89 (1938), ibid. p1340, p1349

L

1

1

W = (1/2) S (xl+1-xl – L)2 -- Springs

V = K S (1-cos 2pxl) -- Substrate

Iff L=1, pseudomorphic

The strength of the coupling to the substrate is given by K. When weak, e.g. large distances, K0, spacing of L. When strong, K inf the spacing will be 1

Displacement z as a function of “n” of xn (extended to continuous).

Solutions in terms of sinc functions, called solitons (which are dislocations by another name)

- These are very rich
- They depend upon both K and L
- Two main cases
- L = N/M (integers), commensurate
- The others (incommensurate)

L=4/5

L=6/7

L=8/9

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=0

V = K S (1-cos 2pxl) -- Substrate

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/50

V = K S (1-cos 2pxl) -- Substrate

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/20

V = K S (1-cos 2pxl) -- Substrate

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/10

V = K S (1-cos 2pxl) -- Substrate

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/5

V = K S (1-cos 2pxl) -- Substrate

L=6/7

Reduce to equivalent positions within 01

In the limit as the repeat period Infinity, all points on curve exist in initial case

L=6/7

L=8/9

- If K is small (weak coupling), all points occupied
- Displacing interface does not change which points are occupied
- Zero static friction (ignoring phonon coupling)

- If K is large enough, strain localized
- Incommensurate set of misfit dislocations (i.e. not periodic)

Unpinned

Zero friction (T=0)

Pinned

K large

T van Erp, PhD thesis, 1999

S. Aubry & P. Y. Ledaeron, Physica D 8, 381 (1983)

+

CSL Boundary Model

Misfit Dislocations

A. Merkle & L. D. Marks, Tribology Letts, 26, 73 (2007)

A. Merkle & L. D. Marks, Phil Mag Letts, 87, 527 (2007)

Friction vs. Misorientation

∑1

∑25

∑13

∑17

∑5

Low energy, low dislocation density, high friction S boundaries.

High friction S orientations not (yet) demonstrated

(Really only Franks’ formula)

S19?

Experiment Theoretical Fit Dominant term is dislocation density

A. Merkle & L.D. Marks, Phil Mag Letts, 87, 527 (2007)

F. Lancon, Europhys. Lett 57, 74, 2002

Displacement z as a function of “n” of xn (extended to continuous).

Solutions in terms of sinc functions, called solitons (which are dislocations by another name)

Dislocation

Displacement as a function of position

Frank-Van der Merwe

Au (111) Cu on Ru

- Juan de la Figuera, Karsten Pohl, Andreas K. Schmid, Norm C. Bartelt
- and Robert Q. Hwang

L can be large (or small), and in 2D problem is richer

Exact match

Near match (would be strained)

Exact match

Strained to match

- The two materials may not exactly superimpose
- No exact CSL
- No exact epitaxy

- Alternative (equivalent) model
- Expand potential in more general form
- Expand elastic strain field
- See paper by Fletcher & Lodge

- To first order in reciprocal space:
- Unitary structure factor
- vo(q) – Interatomic potential term
- k – Distance between diffraction spots (wavevector of elastic distortion) – dominates if small

- Crystal has a periodic potential
- V(r) = S v(g)exp(ig.r)
- Periodic displacements in quasicrystal

- Quasicrystal has an quasiperiodic potential
- W(r) = S w(q)exp(iq.r)
- Quasiperiodic displacements in crystal

Crystal

W(r)

V(r)

Quasicrystal

- (Following Fletcher & Lodge)

Interfacial energy calculation

Ignored Ignored

l=12o

D=6o

Calculated energies for two kTotal energy

Minority

Majority

Widjaja & Marks, Phil Mag Letts, 2003. 83(1) 47.

Widjaja & Marks, PRB, 2003. 68(13) 134211.

- FK (FVdM) models are solvable approximations
- Strain localization/solitons/misfit dislocations
- Commensuration matters
- Commensurate: periodic array of misfit dislocations
- Incommensurate, either aperiodic array of misfit or no matching

- In 2D problem can be more complicated
- Rotated alignments

- Near Coincident orientations
- Energy scales ~1/k, alignment in reciprocal space

S.L. Dudarev, J.-L. Boutard, R. Lässer, M.J. Caturla, P.M. Derlet, M. Fivel,

C.-C. Fu, M.Y. Lavrentiev, L. Malerba, M. Mrovec, D. Nguyen-Manh,

K. Nordlund, M. Perlado, R. Schäublin, H. Van Swygenhoven,

D. Terentyev, J. Wallenius, D. Weygand and F. Willaime

EURATOM Associations

Vacancy Motion Interstitial Motion

The dynamics of microstructural evolution

50 nm

Thermal Brownian motion of nanoscale prismatic dislocation loops in pure iron at 610K (courtesy of K. Arakawa, Osaka University, Japan).

Science 318 (2007) 956

Growth of dislocation loops in ultra-pure iron under in-situ self-ion irradiation at 300K (courtesy of Z. Yao and M. L. Jenkins, Oxford University, UK).

Philos. Magazine (2007) in the press

The fundamental microscopic objects

P. Olsson, 2002

Density functional theory calculations showed that magnetism was responsible for one of the most significant feature of the FeCr phase diagram (2002). DFT calculations also identified the pathways of migration of defects in iron (2004), as illustrated by the movie above.

Migration of radiation defects in pure metals

Fe: migration of a single 110 self-interstitial defect at 200°C.

Fe or W: migration of a 61-atom self-interstitial atom cluster at 200°C.

W: migration of a single 111 self-interstitial defect at 500°C.

Radiation defects produced by collision cascades in pure metals migrate very fast (linear velocities are in the 100 m/s range, and diffusion coefficients are of the order of ~10-9 m2/s).