Bob Pond Acknowledge: John Hirth Classical Model (CM) Geometrical - PowerPoint PPT Presentation

Mechanistic Models of Deformation Twinning and Martensitic Transformations Bob Pond Acknowledge: John Hirth

Classical Model (CM) Geometrical – invariant plane Topological Model (TM) Mechanistic – coherent interfaces, interfacial line-defects

CM TM 𝒄 Ʇ ℎ γ = 𝑐/ℎ 𝑡 Twinning dislocation: e.g. F.C. Frank, 1949 (disconnection) Bilby & Crocker, 1965 Twinning : e.g. G. Friedel, 1926 Martensitic Transformations PTMC : WLR and BM, 1953 Pond and Hirth, 2003

Interfacial defect character and kinetics

Admissible interfacial defects Operation characterising defect (𝑿 𝝁 , 𝒙 𝝁 )(𝑿 𝝂 , 𝒙 𝝂 ) −1 Interfacial dislocations white crystal 𝑱, 𝒄 𝝁 𝒐 Twinning disconnections (𝑿 𝝁 , 𝒙 𝝁 ) ℎ 𝒄 = 𝒖 𝝁 − 𝑸𝒖 𝝂 ℎ = 𝒐 ⋅ 𝒖 𝝁 (𝑿 𝝂 , 𝒙 𝝂 ) γ = 𝑐/ℎ black crystal 𝒄 μ −𝒖 𝝂 𝒖 𝝁 Pond, 1989 bicrystal

Thermally activated disconnections • activation energy at fixed stress ~ 𝑐 2 loop nucleation rate, ሶ 𝑂 , reasonable for small 𝑐  • defect mobility, ሶ 𝐻  enhanced by larger core width, 𝑥 , w hich is promoted by small ℎ  simple shuffles

Motion of a twinning disconnection in a twin 𝒖 𝝁 [10 ത 10] b 𝐹 𝑗 = 0.26 𝐾𝑛 −2 h 𝒖 𝝂 [0001] 𝒄 = 0.062 𝑜𝑛 ℎ = 2𝑒 (10ഥ 12) = 0.376 𝑜𝑛 γ = 𝑐/ℎ 𝛽 − 𝑈𝑗 𝑥~6𝑏 𝐶𝑠𝑏𝑗𝑡𝑏𝑨 𝑓𝑢 𝑏𝑚. 1966 𝑒 = 1 𝑁𝑄𝑏 𝜏 𝑄

Atom Tracking: Shear and Shuffle Displacements in Twin (10ത 12) z 4 distinct atoms y “swapping” “rocking” Pond et al., 2013

Deformation twins in Ni 2 MnGa inter-variant boundary Disconnection 𝒄 = 1 12 10 ത 1 = 0.072 𝑜𝑛 𝒉 = 𝟏𝟏𝟐𝟑 ℎ = 𝑒 (202) = 0.211 𝑜𝑛 𝛿 = 𝑐 ℎ = 0.34 Pond et al. 2012 𝒉 ⋅ 𝒄 = 𝟐 Zarubova et al. 2012

Twin tip in Ni 2 MnGa 𝐹 𝑗 = 0.01 𝐾𝑛 −2 4 distinct atoms no shuffling h ℎ 𝒉 = 𝟑𝟏ഥ 𝟑 Muntifering et al. 2014 SF HAADF STEM (Titan PNNL)

Topological model for type II twinning

Classical Model: irrational plane of shear 1 2 𝛽 𝛽 1 Type I Type II 1 rational 2 irrational 2 φ 2 1 i rrational 2 rational 1 = 2 1 = 1 1 2 φ 2 −𝛽 2 s 2 1 1 s 2 1 + 𝛽 𝑡 = 2𝑢𝑏𝑜𝛽 2 1 1 = 1 1 = 2

TiNi μ 1ത 10 𝜈 000 101 𝜇 𝜇 11ത 1 𝜈 Knowles, 1982 (a) (b)

ሶ Formation mechanisms for type I and II twins Type I: glide twin Type II: glide/rotation twin 1 = 1 𝑡 1 = 2 type I source 𝑂 type II 𝑡 𝐻ሶ type II 2 φ 1 = 1 2 Type I: glide twin source 𝛿 = 𝑐/ℎ 𝑂ሶ 2 φ 𝐻/ ሶ 𝐻ሶ 𝑂 favours type I competitive mechanisms: High ሶ 1 1 = 2 2 𝛽 𝐻/ ሶ 𝑂 favours type II Low ሶ

Type II: formation of glide/rotation twin disconnection glide plane, k 1 1 = 2 h twin parent Ʇ b /2 b /2 b unsheared region sheared region Ʇ 1 = 2 Ʇ Ʇ 𝛽 h 1 (a) Ʇ b g (c) Ʇ parent twin (b)

Type II: growth Ʇ Ʇ 1 = 2 𝛿 = 𝑡 = 2ta (𝛽) Ʇ Ʇ Ʇ 2𝛽 Ʇ Ʇ Ʇ Ʇ b g Ʇ (ii) (ii) (i) (i) Ʇ (b) parent twin Read and Shockley, 1953 1 = 2 (a)

ሶ Experimental observations: e.g. 𝛽 − 𝑉 𝑯/ ሶ 𝟑 𝑐 ℎ 𝛿 No. dist. 𝑂 1 1 type nm nm atoms " {17 ത 111 0.098 0.456 0.216 4 low 6 }" 1/2 < 512 > II 112 0.081 0.356 0.228 4 low " 1 ത 1/2 < 312 > 72 " II 1ത 110 0.048 0.161 0.299 2 high 30 1/2 < 310 > compound 10𝜈𝑛 Type II Twinning in Other Systems NiTi CuAlNi 𝛽 − 𝑉, 𝐷𝑏ℎ𝑜 1953 TiPd devitrite

Topological model of martensitic transformations

PTMC TM • low energy terraces (coherently strained epitaxial) d martensite • two defect arrays: disconnections & LID • distortion field of defect network accommodates coherency strains • motion of all defects produces shape deformation p ’ invariant plane parent Shape deformation P 1 = RBP 2 = ( I + dp ’)

Glissile Disconnections Ti 10 wt % Mo Klenov 2002 𝒄 ℎ 𝜇 ℎ(𝜈 ) 𝒄 • 2 distinct atoms 𝒖 𝜈 • steps cause habit plane to be inclined to terrace plane 𝒖 𝜇 • 𝒄 𝑜 also produces rotational distortions • motion causes one-to-one atomic exchange between phases with different densities 𝒄 𝑜 = ℎ 𝜇 − ℎ(𝜈)

Distortion field of a Defect Array Z’ d h Lagrangian frame y’ X’ b b s b e habit plane ξ

Equilibrium: superposed coherency and defect array distortion fields  D Solve the Frank-Bilby Equation for the defect array with long-range 𝒏 , which compensates 𝑬 𝒋𝒌 𝒅 . distortion matrix, 𝑬 𝒋𝒌

Habit plane orientation Ti : φ = 0.53° β crystal: Θ - φ θ = 11.4 ° homogeneous isotropic α crystal: Θ + φ approximation φ θ inhomogeneous anisotropic case rotations partitioned according to relative elastic compliances TM solutions for habit plane orientation differ slightly from PTMC, unless 𝒄 𝑜 = 0

Partitioning of rotations 𝑑 𝜗 𝑧𝑧 = 12.33% Cu Ag molecular dynamic simulation of static Cu(111)/Ag(111) interface, Wang et al. 2011  Cu  Ag  -  Ag /  Cu Case Isotropic, inhomogeneous 0.449 -0.698 1.15 1.55 Anisotropic 0.504 -0.853 1.36 1.69 MD 0.483 -0.929 1.41 1.92 MD (Artificial) 0.665 -0.659 1.312 0.97

Orthorhombic to Monoclinic Transformation in ZrO 2 considerable shuffling: Principal strains on terrace plane 8 Zr & 16 O distinct ions     0 3 . 8 % xx yy

synchronous motion of disconnections d D habit  terrace y Chen and Chiao, 1985  D     0 0 b     xz x    y     D     0 0 b 0 0 n m yz y z D   d        0 0 b zz z

Lath martensite in ferrous alloys 1: terrace plane “ N- W OR” Mn IF steel: Morito et al.

TEM: LID slip dislocations ~{575} G-T OR dislocations, ~10 ° from screw, with 1 / 2 [ 1 1 1 ]  spacing 2.8 -6.3 nm Fe-20Ni-5Mn (Sandvik and Wayman, 1983)

TEM: Disconnections in near screw orientation Moritani et al. Fe-Ni-Mn [-101] γ projection

Plate Martensite ~{121} Ogawa and Kajiwara, 2004 Fe-Ni-Mn

Conclusions Topological modelling provides insights into mechanisms and kinetics. Twinning:  proposed new model of type II twin formation. Martensite:  predicted interface structures consistent with observations,  predicted habits differ slightly from PTMC.