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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Lattice structure of deformation twins in fcc alloys B 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

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

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

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.

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

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

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

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

Calculated vs Experimental Planar Defect Energies May 2007

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

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

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

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

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

“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-0325939) “He who has imagination without learning has wings but no feet.” Joseph Joubert, essayist (1754-1824) May 2007