Interphase

1. 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

2. 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

3. 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

4. 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

5. 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)

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

7. L=4/5

8. L=6/7

9. L=8/9

10. W = (1/2) S (xl+1-xl – L)2 -- Springs K=0 V = K S (1-cos 2pxl) -- Substrate

11. W = (1/2) S (xl+1-xl – L)2 -- Springs K=1/50 V = K S (1-cos 2pxl) -- Substrate

12. W = (1/2) S (xl+1-xl – L)2 -- Springs K=1/20 V = K S (1-cos 2pxl) -- Substrate

13. W = (1/2) S (xl+1-xl – L)2 -- Springs K=1/10 V = K S (1-cos 2pxl) -- Substrate

14. W = (1/2) S (xl+1-xl – L)2 -- Springs K=1/5 V = K S (1-cos 2pxl) -- Substrate

15. 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

16. 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)

17. 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)

18. + 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)

19. 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)

20. 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)

21. Change in friction above transition F. Lancon, Europhys. Lett 57, 74, 2002

22. 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)

23. Frank-Van der Merwe Dislocation Displacement as a function of position

24. Frank-Van der Merwe

25. 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

26. Role of the Vernier L can be large (or small), and in 2D problem is richer

27. Ö2, 450 rotation

28. Ö2, 450 rotation

29. Hexagonal on Square

30. Hexagonal on Square Exact match Near match (would be strained)

31. Hexagonal on Square Exact match Strained to match

32. 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

33. Sketch of Model 1

34. Sketch of Model 2

35. Sketch of Model 3

36. 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

37. 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)

38. Interfacial energy calculation Ignored Ignored l=12o D=6o Calculated energies for two kTotal energy

39. Experiments + Theory Minority Majority Widjaja & Marks, Phil Mag Letts, 2003. 83(1) 47. Widjaja & Marks, PRB, 2003. 68(13) 134211.

41. 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

42. 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

43. Brownian Motion Vacancy Motion Interstitial Motion

44. 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

45. 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.

46. 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).

47. Carbide Barriers

48. Large Precipitates

49. Stacking Fault Interactions

50. High Temperature