Pattern formation during diffusion limited transformations in solids

1. Pattern formation during diffusion limited transformations in solids

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

3. Diffusional phase transitions interface • Thermal diffusion • Heat conservation • (Local) phase equilibrium The chemical potential depends on the elastic state dimensionless temperature: diffusion constant: capillary length: latent heat:

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

5. Figure 4: Phase field of a growing finger 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 3: Steady state free growth of a bicrystal

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

7. Thermal diffusion Phase field modeling • Free energy functional • Free energy density ( ) • Phase field kinetics • Elastodynamics ( mass density) Figure 5: Double well potential:

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

9. 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 7: Single crystal growth • Figure 8: first order phase transition: • symmetrical- to parity broken finger

10. 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. 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 10: Single crystal growth Figure 11: first order phase transition: symmetrical- to parity broken finger

12. Bicrystal: Free growth • 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 Figure 12: Growth of a bicrystal

13. Bicrystal growth • 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 Figure 13: Growth of a bicrystal • Figure 14: first order phase transition: • single- to twinned bicrystal finger

14. 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