The importance of shear in the bcc hcp transformation in iron

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1 The importance of shear in the bcc hcp transformation in iron Kyle J. Caspersen and Emily Carter Department of Chemistry and Biochemistry University of California Los Angeles Adrian Lew * and Michael Ortiz Graduate School of Aeronautics California Institute of Technology Funded by the Department of Energy - Accelerated Strategic Computing Initiative (DOE-ASCI) * Stanford University

2 Phase transformations in iron 6 Vocadlo, el.al, Faraday Discuss. 16, 25 (1997). Liquid bcc? bct? A strong shock wave will induce phase transitions producing complicated microstructure. 4 T (K) 2 fcc Saxena Boehler dhcp? hcp Hidden variables? bcc 1 2 P (GPa) Ground state ferromagnetic bcc undergoes a martensitic phase transformation to non-magnetic hcp at ~1 GPa. BCC HCP a c Large scatter in the measured TP Fuzzy phase boundaries Mixed states Large hysteresis Goal: understand scatter and hysteresis in transition pressures

3 Phase transformations in iron 6 Vocadlo, el.al, Faraday Discuss. 16, 25 (1997). Liquid bcc? bct? A strong shock wave will induce phase transitions producing complicated microstructure. 4 T (K) 2 fcc Saxena Boehler dhcp? hcp Hidden variables? After Transition bcc BCC 1 2 P (GPa) Ground state ferromagnetic bcc undergoes a martensitic phase transformation to non-magnetic hcp at ~1 GPa. HCP a c Large scatter in the measured TP Fuzzy phase boundaries Mixed states Large hysteresis Goal: understand scatter and hysteresis in transition pressures

4 Martensite in iron length scales 1μm Variants of bcc Fe produced by bcc hcp bcc transformation induced by shock loading (Bowden and Kelly) Micron-length scales require continuum analysis!

5 Atomistic continuum connexion undeformed Cauchy-Born Rule deformed

6 Kinematics of bcc hcp bcc phase transformations <1> <1> compression <11> elongation <11> shuffle Burgers path <11> U u <11> <1> U(V) ( V ) α( V ) α α( V ) α( V ) = u = 4 2 View of <11> Plane initial bcc variant 6 hcp variants 12 bcc variants U is an hcp deformation u is the shuffle V = c ( ) ( a ) V α 8 UG G -1 U -1 HUG G bcc point group H hcp point group 19 total variants

7 Multi-well elastic energy for iron W Assumption: each deformation close to deformation of a variant Ground State hcp i ( F ) = min W ( F ) i=,...,18 DFT calculations prove to costly for on-the-fly W(F) or tabulated W(F) W( F ) Taylor Expansion W i i 1 i ( C) = W ( V) + C C ( V) Γ i ( V) 2 2 i W = 2 C DFT Calculations T i i ( ) Γ ( V) ( C C ( V) ) Variant deformation, C i (V) (hcp c/a ratio) i i i Equation of State, W (V) = W ( C (V)) Non-Linear Elastic Constants, Г i (V) (Calculated using volume conserving shears) C i ( V ) C = F T F bcc and fcc : Г 11 Г 12 Г 44 hcp : Г 11 Г Г 12 Г 1 Г 44 V Ground State bcc Approximation: Taylor expansion around variant deformation C = DFT Details Kohn-Sham DFT within VASP GGA and PW-91 Projector Augmented Wave (PAW) all electron method 2 Ions/Cell Monkhorst Pack K-Point Grid 5 ev Kinetic Energy Cut Off Spin-Polarized for bcc

8 First-principles input into continuum model bcc hcp bcc bcc hcp EXPERIMENT FLAPW* PAW Transition Pressure (GPa) GPa bcc elastic constants compare well with experiment (C 11 a bit high) little experimental data for hcp PAW predicts the bcc to hcp transition pressure within the measured range * Herper, et.al, Phys. Rev. B 6, 89 (1999). Pressure (GPa) bcc

9 Multi-well elastic energy phase mixing! Multi-well free-energy density with 19 variants: V initial V final W( F ) ε f Ground State hcp Prescribed average deformation V,T Ground State bcc Approximation: Taylor expansion around variant deformation C = Free-energy minimizers need not be uniform deformations (pure phases) Free energy can be reduced by mixing different phases (deformation patterning) Goal: understand energyminimizing deformation patterns (microstructures)

10 Microstructure of martensite Cu-Al-Ni, Chunhua Chu and Richard D. James

11 Microstructure of martensite Cu-Al-Ni, Chunhua Chu and Richard D. James

12 Sequential laminates Ansatz: Free-energy minimizing microstructures are sequential laminates Simple laminate Coherent interfaces Must optimize: Interfacial orientations Volume fractions Nesting of laminates Variant sizes Constraints: Equilibrium at interfaces Coherent interfaces Rank-2 laminate (Aubry, Fago and Ortiz, CMAME, 2)

13 Shear Compression Vinitial Linear Transformation 1 λ F (δ ) = (1 δ ) 1 + δ ε f 1 Vfinal undeformed bcc εf ε f =. (Note: det[f]=v) ε f =.4 Note: mapped deformed volume onto reference volume εf λ λ λ is set such that V=Vf

14 Role of Shear.2..4 V Transformation Volume (V\V bcc ) Pressure (GPa) Volume (V/V bcc ) Transformation Pressure (GPa) ε f - shear is required to activate this transformation - increasing shear lowers the TP and increases the TV - variability in measured TPs may be due to shear states

15 bcc hcp Phase Transformation What are the average bulk properties of the bcc hcp transformation? To explore the transformation, we apply a series of volume-conserving deformations (F) along the transformation path. U ( V) = ( V ) α( V ) α 2 α 2 + ( V ) α( V ) ( V) = ( c a ) V α 8 bcc deformation hcp deformation F hcp linear transformation ( V) V 1 Ι F bcc = V detu(v) ( V) = U(V) [ ] ( δ) = ω (1 δ) F ( V) δf ( V) F + V, δ bcc hcp ω = volume-conserving scale factor δ 1 1 δ Volume The F(V,δ) that minimizes W determines the transformation properties.

16 bcc hcp Phase Transformation Energy (ev/ion) Pressure (GPa) Fraction hcp V hcp Volume (Å /Ion) V bcc full conversion to hcp transition pressure of 1 GPa hallmarks of Gibbs construction - the lowering of the energy - the lag to full conversion to hcp deviation from Gibbs construction - no perfect tangent matching - energy is increased - caused by the two imposed constraints Hadamard compatibility condition F - F = a n dependence on the transformation path - constraints introduce frustration 1 hysteresis width of 5.2 GPa observed - loading TP = 1.2 GPa - unloading TP = 5. GPa - experimental width 6.2 GPa (Taylor et al.) 2

17 Directional Deformation of bcc Fe propagating shock waves apply load in specific directions to investigate directional loading we applied the directional deformation F θ,φ (δ) spanning all direction space (angle space) c b a θ φ what effect does the direction of applied load have on the transformation? cosφ sinφ cosθ sinθ ( a, b, c ) = sinφ cosφ 1 ( δ) = δ( a a ) + ( b b ) + ( c c ) F θ,φ 1 a sinθ cosθ

18 Directional Deformation of bcc Fe: TP and TV P (GPa) Transition Pressure Transition Volume φ Degrees 4 φ Degrees Transition Pressure θ Degrees θ Degrees V/V bcc φ Degrees 4 φ Degrees TP for all angles is high, >2 GPa - never optimal combination of 4 Transition Volume contraction and and shear region of high pressure for moderate angles - perhaps no hcp variant along path θ Degrees θ Degrees c b θ a loading parallel to simple facets facilitates the transformation - notably the 11 planes loading along the simple axes recovers smallest TP in that region of angle space < 11 < 1 φ a

19 First-principles input at finite temperature Ks (GPa) bcc Fe P(GPa) K bcc Fe 225 K G (GPa) T (K) P(GPa) (Sha, Cohen, Steinle-Neumann) V/V 1 K hcp Fe K V (bohr)

20 Effect of temperature and plasticity The pressure/volume properties of Fe subject to external strain at room temperature 25 2 K,.6 K,.6 K,.4 K,.2 Preliminary results show that the temperature effects are not very significant, but still the phase transformation tends to occur at lower pressures at room temperature than at zero temperature. Pressure (GPa) 15 1 K results: Lew et al., J. Mech. Phys. Solids 54, 1276 (26). Plasticity also generates lamintes: V/V (Sha, Cohen, Steinle-Neumann) Shocked Ta (Meyers et al., 1995)

21 Conclusions Shear Compression shear is required for transformation to occur increasing shear lowers the TP and increases the TV sensitivity to shear may be responsible for the variability in the measured TPs bcc-to-hcp Transformation full conversion to hcp at 1 GPa, consistent with the experimentally observed values hallmarks of the Gibbs construction deviation from perfect mixing due to imposed constraints shows hysteresis in the TP simply due to the crystalline kinematics Directional Deformation transformation pressure is > 2GPa region of very high transformation pressure loading to simple facets facilitates the transformation, in particular along simple axes = a n Acknowledgements Matt Fago :S. Aubry, M. Fago, and M. Ortiz, Computer Methods in Applied Mechanics and Engineering 192, 282 (2). De-en Jiang Robin Hayes Emily Jarvis, Sha, Cohen, Steinle-Neumann Energy (ev/ion) Pressure (GPa) F 1 Volume (V/V bcc ) Volume (Å /Ion) - F 2 TV (V\V bcc ) Pressure (GPa) ε f Volume (Å /Ion) TP (GPa)