1. Mitglied der Helmholtz-Gemeinschaft E. A. Brener Institut für Festkörperforschung, Pattern formation during diffusion limited transformations in solids

2. Overview ‣ Solid-solid transformations ‣ Numerical methods ‣ Model studies

3. E.Brener, Institut für Festkörperforschung Diffusional phase transitions ‣ Thermal diffusion ‣ Heat conservation ‣ (Local) phase equilibrium dimensionless temperature: diffusion constant: capillary length: latent heat: The chemical potential depends on the elastic state interface

4. E.Brener, Institut für Festkörperforschung ‣ Displacements coherent at interface ‣ Free energy of reference- and new phase (sum convention!) ‣ Eigenstrain: dilatational or shear Solid-solid phase transitions Figure 1: Coherent interface with dilatational eigenstrain Figure 2: Hexagonal to orthorhombic transition Displacement field: Strain tensor: Elastic constants:

5. E.Brener, Institut für Festkörperforschung Treating the moving boundary problem ‣ Free growth: Boundary integral method ‣ closed formulation requires symmetrical model ‣ mapping the interface-strain-jump to force density ‣ Channel growth: Phase field technic ‣ phase field with bulk values ‣ smooth interface with width ‣ solve equations of motion in the hole computational area Figure 4: Phase field of a growing finger Figure 3: Steady state free growth of a bicrystal

6. E.Brener, Institut für Festkörperforschung Boundary integral method Figure 3: Steady state free growth of a bicrystal ‣ Eigenstrain mapped to force density ‣ Integral representation ‣ Elastic hysteresis ‣ Steady state interface equation ‣ : Control prameter ; Driving force ; Eigenvalue Peclet number ; modified Bessel function

7. E.Brener, Institut für Festkörperforschung Phase field modeling ‣ Free energy functional ‣ Free energy density ( ) ‣ Phase field kinetics ‣ Elastodynamics ( mass density) ‣ Thermal diffusion Figure 5: Double well potential:

8. E.Brener, Institut für Festkörperforschung Channel growth ‣ Elastic hystereses shift ‣ Heat conservation ‣ Critical phase fraction Figure 6: Single crystal and bicrystal setup Strength of elastic effects: Type of eigenstrain: Thermal insulation - fixed displ. Thermal insulation - stress free

9. E.Brener, Institut für Festkörperforschung Dilatational eigenstrain ‣ No steady state solution in free space ‣ Found two different steady state patterns in finite channel ‣ Symmetrical finger ‣ Parity broken finger ‣ Velocity selection by the channel ‣ Figure 8: first order phase transition: symmetrical- to parity broken finger Figure 7: Single crystal growth

10. E.Brener, Institut für Festkörperforschung Single crystal: Free growth ‣ Mixed mode eigenstrains ‣ Found steady state solution in free space ‣ Velocity selection by elasticity is much more effective then by e.g. anisotropy ‣ Elasticity ‣ Anisotropy Figure 9: Single crystal free growth results

11. E.Brener, Institut für Festkörperforschung Single crystal: Channel growth ‣ Eigenstrain orthogonal to the growth direction: ‣ Velocity selection by elasticity much more effective then by the channel ‣ Good quantitative agreement between the two methods ➡ Phase field confirms dynamic stability of the BI- solution ‣ Figure 11: first order phase transition: symmetrical- to parity broken finger Figure 10: Single crystal growth

12. E.Brener, Institut für Festkörperforschung Bicrystal: Free growth Figure 12: Growth of a bicrystal ‣ Hexagonal to orthorhombic transformation ‣ Found dendrite-like bicrystal solution in free space ‣ found also solution with a „week triple junction“ ➡ Selection by elasticity ‣ Recover bicrystal with phase field method Reminder: Hexagonal to orthorhombic transition

13. E.Brener, Institut für Festkörperforschung Hexagonal to orthorhombic transformations Y. H. Wen, Y. Wang, L. A. Bendersky and L. Q. Chen, Acta Mater. 48, 4125 (2000)

14. E.Brener, Institut für Festkörperforschung Hexagonal to orthorhombic transformations: Strain influence Y. H. Wen, Y. Wang, and L. Q. Chen, Acta Mater. 49, 13 (2001)

15. E.Brener, Institut für Festkörperforschung Bicrystal growth Figure 13: Growth of a bicrystal Figure 14: first order phase transition: single- to twinned bicrystal finger ‣ Found dendrite-like bicrystal solution in free space (by boundary integral technic) ‣ Recover bicrystal with phase field method ➡ Indication of a dynamically stable solution ‣ For shear eigenstrain with 10% dilatation, found transition to twinned finger ‣ Comparison of growth velocities shows very nice agreement

16. Conclusion ‣ Solid-solid transformations ‣ Elastic effects ‣ Diffusional phase transitions ‣ Two complementary methods ‣ Free growth: boundary integral ‣ Channel growth: Phase field ‣ Model study ‣ Single crystal ‣ Bicrystal