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Interphase - PowerPoint PPT Presentation

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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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Low angle


High angle


Based on angle of rotation




Based on axis


Based on Lattice Models




Based on Geometryof the Boundary plane


Wulff-type constructions



  • 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

frankel kontorova model




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)

W = (1/2) S (xl+1-xl – L)2 -- Springs


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


W = (1/2) S (xl+1-xl – L)2 -- Springs


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


W = (1/2) S (xl+1-xl – L)2 -- Springs


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


W = (1/2) S (xl+1-xl – L)2 -- Springs


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


W = (1/2) S (xl+1-xl – L)2 -- Springs


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

incommensurate case
Incommensurate Case


Reduce to equivalent positions within 01

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



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


Zero friction (T=0)


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)


Friction vs. Misorientation






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


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


Displacement as a function of position

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

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
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





  • (Following Fletcher & Lodge)

Interfacial energy calculation

Ignored Ignored



Calculated energies for two kTotal energy

experiments theory
Experiments + Theory



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
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


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).