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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 Geometry of the Boundary plane. Curved. Wulff-type constructions. Faceted. Mixed.

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Interphase

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Interphase

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


Interfaces

Interfaces

  • 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


Interfaces1

Interface Bonding

Interfaces

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


Frankel kontorova model

L

1

1

Frankel Kontorova Model

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, K0, spacing of L. When strong, K inf the spacing will be 1


Frank van der merwe

Frank Van der Merwe

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)


Fk solutions

FK Solutions

  • These are very rich

  • They depend upon both K and L

  • Two main cases

    • L = N/M (integers), commensurate

    • The others (incommensurate)


Interphase

L=4/5


Interphase

L=6/7


Interphase

L=8/9


Interphase

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=0

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


Interphase

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/50

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


Interphase

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/20

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


Interphase

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/10

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


Interphase

W = (1/2) S (xl+1-xl – L)2 -- Springs

K=1/5

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


Incommensurate case

Incommensurate Case

L=6/7

Reduce to equivalent positions within 01

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

L=6/7

L=8/9


Aubry transition

Aubry Transition

  • 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)


Aubry transition with k

Aubry Transition with K

Unpinned

Zero friction (T=0)

Pinned

K large

T van Erp, PhD thesis, 1999

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


Sliding is dislocation motion

+

Sliding is dislocation motion

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)


Interphase

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)


Sliding on graphite comparison of theory experiment

Sliding on Graphite: Comparison of Theory & Experiment

S19?

Experiment Theoretical Fit Dominant term is dislocation density

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


Change in friction above transition

Change in friction above transition

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


Frank van der merwe1

Frank Van der Merwe

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)


Frank van der merwe2

Frank-Van der Merwe

Dislocation

Displacement as a function of position


Interphase

Frank-Van der Merwe


Examples of solitons stm

Examples of Solitons (STM)

Au (111) Cu on Ru

  • Juan de la Figuera, Karsten Pohl, Andreas K. Schmid, Norm C. Bartelt

  • and Robert Q. Hwang


Role of the vernier

Role of the Vernier

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


2 45 0 rotation

Ö2, 450 rotation


2 45 0 rotation1

Ö2, 450 rotation


Hexagonal on square

Hexagonal on Square


Hexagonal on square1

Hexagonal on Square

Exact match

Near match (would be strained)


Hexagonal on square2

Hexagonal on Square

Exact match

Strained to match


Near coincidence

Near Coincidence

  • 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


Sketch of model 1

Sketch of Model 1


Sketch of model 2

Sketch of Model 2


Sketch of model 3

Sketch of Model 3


Interface orientation

Interface orientation

  • 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


Bring two surfaces into contact

Bring two surfaces into contact

  • 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)


Interphase

Interfacial energy calculation

Ignored Ignored

l=12o

D=6o

Calculated energies for two kTotal energy


Experiments theory

Experiments + Theory

Minority

Majority

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

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


Summary

Summary

  • 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


Brownian motion of defects

Brownian Motion of Defects

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


Brownian motion

Brownian Motion

Vacancy Motion Interstitial Motion


Interphase

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


Interphase

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.


Interphase

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).


Carbide barriers

Carbide Barriers


Large precipitates

Large Precipitates


Stacking fault interactions

Stacking Fault Interactions


High temperature

High Temperature


Frank rhead source

Frank-Rhead Source


Dislocation source 1

Dislocation Source 1


Dislocation source 2

Dislocation Source 2


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