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Mitg
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Hel
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aft
E. A. Brener
Institut für Festkörperforschung,
Pattern formation during diffusion limited transformations in solids
Overview
‣ Solid-solid transformations
‣ Numerical methods
‣ Model studies
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
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:
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
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 ; EigenvaluePeclet number ; modified Bessel function
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:
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 Thermal insulation - fixed displ.displ.
Thermal insulation - fixed Thermal insulation - fixed displ.displ.
Thermal insulation - stress Thermal insulation - stress freefree
Thermal insulation - stress Thermal insulation - stress freefree
E.Brener, Institut für Festkörperforschung
QuickTime™ and a decompressor
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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
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
E.Brener, Institut für Festkörperforschung
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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
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
E.Brener, Institut für Festkörperforschung
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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
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
