1. Department of Materials Science and Engineering
From Schrödinger’s Equation to Rolling Mill: Joining Complex Process at the Micro and Meso Scales to Predict Onset Twinning Stress
Duane D. Johnson
Department of Materials Science and Engineering
May 2007

2. Motivation Outline Deformation in Solids: Shearing versus Twinning
Twinning is observed in many engineering, chemical and geological materials. Yet, unlike shearing, there is no “stress” criterion for twinning.
First-Principles Density-Functional Theory calculations provides system-dependent potential energy surfaces arising from deformation and creation of atomic-scale defects.
Dislocation mechanics describes well the energetics of dislocation interactions outside the “dislocation core”, which required atomistic simulations.
Can we integrated the methods to predict quantitatively critical twinning stress?
Deformation in Solids: Shearing versus Twinning
– Shearing changes shape, twinning changes symmetry.
How does twinning occur? Move dislocations, create planar defects:
– Thousands of atoms needed to simulate these processes directly.
– “dirt” matters
Integrate complex defect interaction to predict twinning stress.
– Requires dislocation activation of twin nucleus, and balance of external applied stress (or energy balance equations).
Coupled Atomic and Mesoscale Models
Outline
May 2007

3. Background Twinning is the primary deformation mechanism in
semiconductor materials like Si (1)
rocks and minerals such as quartz and silicates (2).
hex. closed-packed (hcp) metals (e.g. Ti, Zn) due to limited slip systems.
The discovery of twinning in calcites, for example, has significantly advanced geologists’ understanding of deformation processes in the Earth’s mantle.
Twinning is also a predominant deformation mechanism in face-centered-cubic (fcc) metals and alloys with low stacking-fault energies, especially at low temperatures where thermally activated cross-slip is inhibited (3).
But, there is no theory to predict the onset critical stress for twinning.
May 2007

4. Principle Deformations: Shearing and Twinning
Crystalline Metal
macroscale 
Shearing
shape change
G~ GPa
e.g. hcp Zn
Twinning
No shape change
Twin Boundary
May 2007

5. Dislocations slip planes incrementally…create defects
Dislocation motion requires the successive movement of a half plane of atoms which create planar faults (since moving on plane).
• Bonds across slipping planes are broken and remade in succession.
From Callister: Fundamentals of Materials Sci. & Eng.
(Courtesy P.M. Anderson)
Long-range Stress Fields
Inch worm or ruck-in-a-rug moving analogy
May 2007

6. Dislocations have barriers to motions,
even thermal activation at low temperatures.
In BCC, for example, corrugated surface represents the energy for the moving dislocations, which tend to lie in the minima. After a double kink pair (blue) nucleates,the kinks migrate (black) so that the dislocation moves from one minimum to another.
Dislocation motion
Kink motion
VKink > VDisl
Perspective, D. Chrzan, Science 310 (2005)
May 2007

7. FCC with Stack Fault (HCP ribbon)
Stacking Faults….mistakes in the stacking sequences
All defects cost energy (mJ/m2 or erg/cm2).
FCC
Face-Centered Cubic
HCP
Hexagonal Closed Packed
FCC
…ABCABCABC…
FCC with Stack Fault (HCP ribbon)
...ABCABABCABC...
May 2007

8. Partial Dislocations Create Stacking Faults and Twins
(111) fcc plane b
Motion of partials
Separation of partials
Partial Dislocations b = b1 + b2
In FCC, due to …ABC… stacking, if partials form, edge repulsion wins out, which creates stacking faulted region in between.
Green Partials Separate. A B C
FCC
HCP
partial
Partial dislocations move apart leave hcp SF ribbon.
ABC = 3 layers
AB = 2 layers
ABCABC converts to ABABAB
May 2007

9. Stacking Faults, Twins Stacking Faults,
Grain Boundaries, Dislocations, …
Dislocation in Ti alloy
magnification~50,000 x
Defects in Polymers
Slip bands and GB
Grain
Boundary SF
Polymer beads
May 2007

10. Yield Stress (YS): deviation from elastic to plastic behavior
Plastic flow (dislocation move) when RSS reaches a material-dependent critical called Critical Resolve Shear Stress, when m RSS = ys (Schmid’s “law”).
Stress (Mpa)
strain
elastic
region n
slip f
 is angle to slip dir.
No such “yield-stress criterion” for Deformation Twinning Stress!
May 2007

11. Length Scale of Lattice Imperfections
Vacancies,
impurities
dislocations
Grain and twin
boundaries
Voids
Inclusions
precipitates
From Chawla and Meyers
May 2007

12. Example Defect Formation from Shearing (FCC metal)
g (hcp) A B A B C
a (fcc)
hcp-fcc energy difference
Ehcp-fcc = Ef [hcp] Ef [fcc] SF 
Shear
Stacking Fault Energy (SFE):
Ef [SF] - Ef [fcc]
Partial dislocations repel making SF
gSF = A
Bond-Counting “Rule of Thumb”:
(*Generally does work for alloys)
Hirth and Lothe, Theory of Dislocations (1968)
May 2007

13. Generalized stacking fault energy (GSFE)
Calculating the “Shear Surface” or GSFE A B C
fcc A B C
isf A B C
maximum
unstable
usf m B
isf uz s C B A
<112> u m
Generalized stacking fault energy (GSFE)
May 2007

14. Require at least 3 layers to define a twin fault energy (on the GSFE)
Shearing on Subsequent Planes yields ESF and Twins
Require at least 3 layers to define a twin fault energy (on the GSFE)
May 2007

15. DFT-derived General Stacking Fault Energies
From experiment:
Suggest that
The twinning stress should depend upon barriers to nucleate a twin, even if the stable twin energy tsf is much less! Why not?
May 2007

16. Lattice structure of deformation twins in fcc alloysB C A
<112>
<111>
fcc
twin
Deformation twin in Hadfield steel [001] orientation 3% strain
I. Karaman, H. Sehitoglu, K. Gall, Y. I. Chumlyakov and H. J. Maier, Acta Mater.(2000).
May 2007

17. Discrete lattice effects
Deformation Processes Span Multiple Length Scales 1Å
1mm
1nm
Discrete lattice effects
Molecular Dynamics
Mesoscale
Continuum
1 ps
1 ns
1 ms
1 s
Length scale
time
Quantum Mechanics
How do we combine the need dislocation array and stress fields with the various planar defects that nucleate into single theory?
May 2007

18. Our Approach: Analytic with reasonable choice
twin nucleation configuration
Mesoscale
e.g.: Peierls-Nabarro Model
Continuum
Discrete lattice effects 1Å
1mm
1nm
1 ps
1 ns
1 ms
1 s
Length scale
time
ab initio DFT
e.g.: Crystal plasticity
May 2007

19. Sequential Multiscale Method Done Analytically
ab initio DFT
Mesoscale
Mesoscale
Generalized stacking fault energy (GSFE)
Continuum
Peierls-Nabarro Model
stacking fault width
Planar fault energy
Mahajan and Chin (1973) Nucleation Model
dislocation array + 3 layer twin nucleus
Predict Deformation Twinning Stress
atomic shear
May 2007
Other nucleation models (e.g., Friedel, Venables) but M&C allows analytic eqs.

20. Twin Nucleus Model (Dislocations + Faults)
Only Ad and Cd are mixed partial dislocations.
All dB twinning partials are pure edge dislocations.
Suggested 3 layer twin nucleus [Mahajan and Chin, Acta Metall. 21 (1973) 1353] .
Our GPFE convergence trends are in agreement with above config.
May 2007

21. Total energy of the twin:
Energy of Twin Nucleation Model Using GSFE
Total energy of the twin:
Wanted:
Predict needed
applied
shear
stress
Critical twin size and twinning stress can be determined by minimizing Etotal relative to d and N.
(N is number of layer of twin nucleus, here N=3)
May 2007

22. Twin Stress Equation based on DFT-GSFE
Relation between twin size and stress based on present analysis
(based on minimizing E vs d:
For large enough unstable TSF
Require only calculated GSFE results, which we get by DFT.
May 2007

23. DFT-derived General Stacking Fault Energies
From experiment:
Suggest that
Our theory:
Suggest that
Results found using VASP (Vienna Atomic Simulation Package)
Plane-wave energy cutoffs and k-meshes set to ensure ±1 meV convergence.
May 2007

24. Calculated vs Experimental Planar Defect Energies
May 2007

25. Calculated vs Experimental Critical Twin Stress
Expt. Suggest relation
Clearly not linear!
Indeed, both theory and experiment agree, for the first time!
But, what does theory suggest?
May 2007

26. Calculated vs Experimental Critical Twin Stress
Analytic estimate of
Ideal twin stress (GPa)!
No dislocations, so 2-4 orders too large !
May 2007

27. Present Theory of Combined Dislocation Mechanics
and DFT-derived GSFE
Theory predicts:
Holds away from 0
Indeed, theory and experiment are ~linear in unstable TSFE.
Experiment cannot determine the unstable TSFE.
May 2007

28. Present Theory Holds For Alloys, too!
Linear slope not the same as elements due to alloying effects.
To lower free energy, solutes diffuse to faults (Suzuki effect) causing the fault energy to decrease, or away from faults to prevent increase.
This cause many interesting observed results, including rapid precipitation, and, here, change of twinning stress.
May 2007

29. Conclusions
Theory provided new insight into physics of twinning because both the dislocaiton array and planar defects where accounted for in system-specific way.(Theory useful)
Theory provides means to predict onset twinning stress, by combing large-scale defects in complex interactions in simple energy minimization problem.
Unstable TSFE is related to nucleation barrier, although experiment cannot determine it. (Theory useful)
Large-scale defect simulations with millions of atoms, or large-scale hierarchical methods just unnecessary for certain types of macroscale predictions, as long as essential physics is accounted for.
Avoids dislocation dynamic simulations (such as running on Blue Gene with 30,000 processors).
May 2007

30. “He who has imagination without learning has wings but no feet.”
Acknowledgements
Thanks to you, and ….
Collaborators
Sandeep Kibey (student, MechSE), Jianbo Liu (post-doc, MSE), Huseyin Sehitoglu (MechSE)
FUNDING
The National Science Foundation
Johnson, Sehitoglu, the Center for Process Simulations and Design
Computing resources made possible through the Materials
Computation Center at UIUC from grant NSF ITR (DMR )
“He who has imagination without learning has wings but no feet.”
Joseph Joubert, essayist ( )
May 2007