Micromechanics of crystalline interfaces - TAM 524: Micromechanics of Materials Spring 2021 Ahmed Sameer Khan Mohammed (Sameer) 5th year graduate ...

Micromechanics of crystalline interfaces

 TAM 524: Micromechanics of Materials
 Spring 2021

 Ahmed Sameer Khan Mohammed (Sameer)
 5th year graduate student (MechSE)
Research: Interfaces in phase-transforming materials, carbon-composites

Topics

I. Frank-Bilby equation

II. Topological Modeling – Relevant to current research in our group
 (Mohammed and Sehitoglu, 2020)

III. A glimpse of “Evolving Interface Theory” – Current research in our group
 (Mohammed and Sehitoglu, 2021)

Nature of Interfaces
 Criterion of classification: Atomic-structure/bonding across the interface
 (Olson-Cohen-Bonnet approach)
 “…one for which
 corresponding lattice planes
 and lines are continuous
 across the interface”)
 Incoherent
 Coherent Semi-coherent

 (Olson an Cohen, 1979)

*Olson, G. B., and Morris Cohen. "Interphase-boundary dislocations and the concept of coherency." Acta Metallurgica 27, no. 12 (1979): 1907-1918

Nature of Interfaces: Real examples

 Criterion of classification: Atomic-structure/bonding across the interface
 (Olson-Cohen-Bonnet approach)
 Incoherent
 Coherent
 Semi-coherent

 T

 Nb on Al2O3
 (mismatch 2%)
 Gutekunst et al. 1997)
 Nb on Al2O3
 (mismatch 12 %) (Xu et al. 2020)

 Gutekunst et al. 1997) Study on Sm-doped Cerium-oxide (focus
Gutekunst, G., J. Mayer, V. Vitek, and M. Rühle, Philosophical Magazine A, (1997)
Gutekunst, G., J. Mayer, V. Vitek, and M. Rühle, Philosophical Magazine A, (1997) on conductivity across the GB)
Xu, Xin, Yuzi Liu, Jie Wang, Dieter Isheim, Vinayak P. Dravid, Charudatta Phatak, and Sossina M. Haile, Nature materials (2020)

Nature of Interfaces: Misfit strain

 Relaxed state Coherent state
 dl
 l lc dc
 dc - dl
 el =
 dc
 Misfit strains in
 individual phases
 d µ - dc
 eµ =
 dc
 µ dµ µc dc

 d µ - dl (if l is taken as reference)
Often, one of the phases is taken as reference: e=
 dl
 (Total misfit strain)

Relieving Misfit strains

 Coherent Interface Semi-coherent Interface

Self-equilibriating field relieves the misfit strain Interface dislocations relieves the coherence/misfit strain
 Also called misfit dislocations

 Ma, Xiao, PhD thesis, 2008

I:Frank-Bilby equation
Calculate the interface defect density required to relieve misfit strains, the spacing of misfit dislocations

 Relaxed/Free state of both phases Coherent state of both phases

 -1
 µD r
 -1
 lD r

 µ Dr
 l Dr

 B= - ( l Dr-1 - µ Dr-1 ) v r
 Burgers vector density
 (Determined in reference frame of µ)
 Ma, Xiao, PhD thesis, 2008

Frank-Bilby equation: Example (Sutton & Ballufi, 1994)

 3 NiO NiO
 ( 001) NiO
 1
 2 Pt Pt
 ( 001)Pt
 Epitaxial thin films (Reference state: Pt)
 Deposited at high temperature
 At 1200 C
 ( 001)Pt ( 001) NiO
 aNiO = 0.424 nm
 [010] aPt = 0.397 nm
 [100]
 aNiO - aPt
 e11 = e 22 = e = » 0.07
 éë 110 ùû [110] aPt
 aPt
 aPt aNiO Uniform Biaxial strain
 aNiO Will be the same with any in-plane rotation

(Sutton, A. P., & Ballufi, R. W., Interfaces in Crystalline Materials, 1994)

NiO Frank-Bilby equation: Example (Sutton & Ballufi, 1994)

 ! 1 ! 1
 3
 Pt b1 = [110]; b2 = éë1 10 ùû
 2 2
 1 (Lattice dislocations of Pt)
2
 æ 1 ö
 ç 1+ e 0 0÷ [001]
 ç ÷ æ1 + e 0 0ö
 ç 0 1
 0÷ ç 1+ e 0 ÷÷ eˆ1 = [110]
 NiO D Pt = ç 0
 -1
NiO D Pt = eˆ2 = éë1 10 ùû
 ç 1+ e ÷
 ç 0 0 1÷ ç 0 0 1 ÷ø
 çç ÷÷ è
 è ø
 æ1 - e 0 0ö Pt D Pt = I d2
 d1
 » çç 0 1 - e 0 ÷÷ ! !
 ç 0 1 ÷ø
 è 0
 NiO
 ( I - NiO DPt-1 ) eˆ1 = B1 = db1 e=
 b1

 D =I-1 1 d1
 Pt Pt ! d1 = d 2 = 4.3 nm
 !
 Pt ( I - NiO DPt-1 ) eˆ2 = B2 = db2 b2
 2 e=
 d2
 (Reference state: Pt)
 (Sutton, A. P., & Ballufi, R. W., Interfaces in Crystalline Materials, 1994)

Interfacial dislocations: Disconnections

 Semi-coherent Interface

 T

Interfacial dislocations in-plane
 No step-character Interfacial dislocations with step character
 Disconnections
 Relevant in Twinning and Phase-Transformations

Interfacial dislocations: Disconnections

 {111}
 T
{111}
 T T

 {r 1 1}

 r – An irrational number

Example 2: A semi-coherent twin boundary
• Twin boundary: Homo-phase interface, why would there be a mismatch??
• Common-perception of twins
 B19¢ NiTi (martensite)
 [001]

 Ti
 
 [010] Ti
 Ni
 TB (Liu and Xie 2003)
 [100]
 (Knowles and Smith 1982)
 (Liu and Xie 2003, 2004)

 • 011 Type II TB “…composed of rational
 {111} FCC e.g. Cu (1 1 1) ledges…”
 TB structure was unclear!
• Completely coherent

 (Mohammed and Sehitoglu, 2020) Proposed a semi-coherent boundary as
 the solution
 (Current research)
 12
ASK Mohammed, H Sehitoglu, Acta Materialia, 2020

{111} Terrace Coherence: Misfit Strains
 B19¢ Unit
 n cell (lattice constants)
 ( 1 11) A [21 1]A [2 11]B DA
 [21 1]A b ¹ 900 n( 1 11)A [21 1]
 A
 [1 11]A c = 4.606 A
 o
 ~1$ M A ( B19¢)
 Coherence Strain
 [001] b = 93.41o g A = -g B » 0.96% [011]
 [011] [011]
 n(11 1 )B
 [010] [100
 n(11 1 ) B] o Cause: Distorted Martensitic M B ( B19¢)
 [11 1]B
 [11 1]
 o
 a = 2.699 [2
 A 11]B [2 11]B
 b = 4.386 AB Unit cell
 Non-Cubic and b ¹ 900 [011]
 [011] DB

 ææ11 g0A 00öö v
 D ¹ 0 ) == ççç00
 DAA ( g AA = 1 00÷÷÷ M A ( B19¢) Frank-Bilby Equation
 ç ÷
 çç00
 èè 0 11÷ø÷ø B=b
 b d B = -(D-1A - D-1B )v
Coherent strains are “relieved” away from d
interface by an interface dislocation array {"""} Linear
 M B ( B19¢) density of Probe
 vector
 Defects
 011 211 13

Semi-coherent twin boundary

 • Need a “admissible” interface
 dislocation {"""}
 • A “crystallographically”
 admissible Burgers vector 011 211
 M A ( B19¢)
 MA
 Twinning Partial!

 MB
 Twin
 Twin
 M B ( B19¢)
 T

 Parent
 Matrix
 b

 Plays a role in Twin growth or • Twin partial is a screw dislocation
 Twin Boundary Propagation • Relieves the interface shear strain away
 • 011 Type II Twin in NiTi from the interface
 1 • Twin variants “recover” from strain farther
 b = [ 011]
 9 from interface
(Ezaz and Sehitoglu 2011)

II: Topological Modeling

 æ 0 -0.0192 -0.0011ö
 æ 0 -0.0192 0 ö
 -(D - D ) = çç 0
 -1 -1
 0 0 ÷÷
 -(D A - DB ) = çç 0
 -1 -1
 0 0 ÷÷ A B
 ç0 0 ÷ø
 v ç0
 è 0 0 ÷ø è 0
 %
 111 !
 %
 {"""} {111}111 !
 d0 {"""}
 d1 K1

 [011] 211 211
 [011]

 0
 d ( A) K1
 36.798 (1 1 1) B

 36.858 (0.8619 1.0229 1.0269) B

 36.858 (0.8619 1.0229 1.0269) B

 Predicted Irrational Indices,
 matching experiment
ASK Mohammed, H Sehitoglu, Acta Materialia, 2020 (section 2.5) 15

III: Evolving Interface Theory

 • Is there an influence of external strain??
 • Any dependence on volume fraction?
 • Does the dislocation-spacing change with
 external strain?
 Topological
 Modeling can
 predict this!

 æhö
 q = tan -1 ç ÷
 è ød

(Mohammed and Sehitoglu, Acta Materialia, 2021)

III: Evolving Interface Theory
 (Mohammed and Sehitoglu, Acta Materialia, 2021)

 External Stimuli
 Strain γ(%) Volume
 Fractions
 Terrace Coherence (Displacement gradient jump)
 Evolving f Î [ 0,1]
 Entities
Dislocation- Dislocation-
 Density Traction Continuity on Terrace Plane
 Spacing
 d r ! 1d
 TB Identity
 ( hkl )
 Average Microstructural Strain
 Energy Minimization

Topics

I. Frank-Bilby equation

II. Topological Modeling – Relevant to current research in our group
 (Mohammed and Sehitoglu, 2020)

III. A glimpse of “Evolving Interface Theory” – Current research in our group
 (Mohammed and Sehitoglu, 2021)

At the end of this lecture,

 • Remember 2 methods to analyze interfaces from a mechanics standpoint

 • Understand underlying principles – crystallographic / stress / strain (mechanics) – on
 which these methods are built

 • Be able to apply them to crystalline interfaces: To what scope?
 o Communication - ask relevant questions to define the problem:
 What is the nature of the interface? What is the need for
 mechanics?
 o Analysis – Formulate how you would apply any of these methods
 for the defined problem

 The Bloom’s taxonomy of learning!
https://cft.vanderbilt.edu/guides-sub-pages/blooms-taxonomy/#:~:text=Familiarly%20known%20as%20Bloom's%20Taxonomy,Analysis%2C%20Synthesis%2C%20and%20Evaluation.

References

• Mohammed, Ahmed Sameer Khan, and Huseyin Sehitoglu. "Modeling the interface structure of type II twin boundary in B19′ NiTi
 from an atomistic and topological standpoint." Acta Materialia 183 (2020): 93-109.
• Mohammed, Ahmed Sameer Khan, and Huseyin Sehitoglu. "Strain-sensitive topological evolution of twin interfaces." Acta Materialia
 208 (2021): 116716.
• Ezaz, Tawhid, and Huseyin Sehitoglu. "Type II detwinning in NiTi." Applied Physics Letters 98, no. 14 (2011): 141906.
• Sutton, Adrian P., Ballufi, R. W., "Interfaces in crystalline materials." Monographs on the Physice and Chemistry of Materials (1995):
 414-423.
• Gutekunst, G., J. Mayer, and M. Rühle. "Atomic structure of epitaxial Nb-Al2O3 interfaces I. Coherent regions." Philosophical
 magazine A 75, no. 5 (1997): 1329-1355.
• Gutekunst, G., J. Mayer, V. Vitek, and M. Rühle. "Atomic structure of epitaxial Nb-Al2O3 interfaces II. Misfit dislocations."
 Philosophical Magazine A 75, no. 5 (1997): 1357-1382.
• Xu, Xin, Yuzi Liu, Jie Wang, Dieter Isheim, Vinayak P. Dravid, Charudatta Phatak, and Sossina M. Haile. "Variability and origins of
 grain boundary electric potential detected by electron holography and atom-probe tomography." Nature materials 19, no. 8 (2020): 887-
 893.
• Olson, G. B., and Morris Cohen. "Interphase-boundary dislocations and the concept of coherency." Acta Metallurgica 27, no. 12
 (1979): 1907-1918
• Ma, Xiao, “Topological Modelling of Martensitic Transformations in Crystalline Materials”, PhD thesis, University of Liverpool, 2008

Thank you!