1. GBs, quick summary so far… Types –Low angle (dislocations from strain localization) –High angle CSL boundaries (low energy) –CSL dislocations Structural unit boundaries (low energy) Low index plane boundaries (low energy)

2. But This only addresses energy versus tilt/twist, what about plane?

4. Wulff Construction The standard approach is to consider how a property scales as a function of the size of the system (R). In a generic sense one can write: P(R) = AR 3 + BR 2 + CR + D A => Bulk behavior B => Surface/Interface term, or at least what scales as a surface term C => Edge/Line term D => Limit for atomic behavior The bulk properties of a material only depend upon “A”; but we have additional terms.

5. Example If P(R) is a free energy B = surface free energy (for a surface); interfacial free energy, grain boundary free energy or stacking fault free energy. Normally use  C = dislocation free energy (line) D = point defect free energy (zero dimension) P(R) is an entropy – similar Other things as well. For instance dE/de (e a strain) is the stress in the bulk. Similarly we can discuss can write d  /de as an interfacial/surface stress term, or a line stress term for a dislocation.

6. Method In the west, proof is generally attributed to Conyers Herring, but a more correct attribution is to Von Laue during the 2 nd world war Max Von Laue Conyers Herring

7. Approach Write the problem as minimizing the total surface free energy as a function of what surface facets are present, for constant volume: Minimize –F =   i M i - (1/3)  m i M i –Note: Lagrangian Solution –  i = m i

8. From  -plot to EQUILIBRIUM SHAPE OF CRYSTAL → the Wulff construction  Draw radius vectors from the origin to intersect the Wulff plot (OA in Figure)  Draw lines  to OA at A (line XY)  The figure formed by the inner envelope of all the perpendiculars is the equilibrium shape

9. Example

10. Gold Octahedra Polyol synthesis developed by Oh Cho group Synthesized by Mirkin group {111} capped, single crystal C. Li, et al., ACS Nano. 2, 1760 (2008)

11. Gold and Silver Cubes Au

12. Crystal shape of pure Cu and of Bi-saturated Cu at ~ 900°C (with monolayer of adsorbed Bi at the surface) illustrates effects of segregation on ECS Cu Bi-saturated Cu Curtesy Paul Wynblatt

13. Example: scanning electron microscope image of a Bi-saturated Cu "negative" crystal Curtesy Paul Wynblatt

14. Morphology of Pb crystals as a function of T Facets Curtesy Andrew Zangwill

15. {110} facet stabilization: cubo-octahedral shape. SrTiO 3 cubes

16. Wulff & Winterbottom 16 γ 100 γ 111 001 110 100 001 γ 111 γ 111 √(3/2) γ Int – γ Sub = 0 0 < γ Int – γ Sub < γ Pt γ Int – γ Sub ≤ -γ Pt -γ Pt < γ Int – γ Sub < 0 γ Int – γ Sub = γ Pt Increasing γ int Increasing γ sub Increasing γ Pt

17. Modified Wulff Construction (twins)

18. Kinetic Wulff construction If, instead of the surface/interface free energy we use growth velocity, a quasi- stationary kinetic shape is generated by exactly the same construction  Often the case when kinetics dominate