Wulff Construction

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)

But

• This only addresses energy versus tilt/twist, what about plane?

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) = AR3 + BR2 + 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.

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.

Method

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

Max Von Laue Conyers Herring

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 = iMi - (1/3) miMi

– Note: Lagrangian

• Solution– i = mi

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

Example

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)

VertexAndEdgeTruncationsOfThePlatonicSolids.nbp

Gold and Silver Cubes

Au


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

Example: scanning electron microscope image of a Bi-saturated Cu "negative" crystal

Curtesy Paul Wynblatt

Morphology of Pb crystals as a function of T

Facets

Curtesy Andrew Zangwill

• {110} facet stabilization: cubo-octahedral shape.

SrTiO3 cubes


Wulff & Winterbottom

16

γ100

γ111

001

110100

001

γ111γ111√(3/2)

γInt – γSub = 00 < γInt – γSub < γPt γInt – γSub ≤ -γPt-γPt < γInt – γSub < 0γInt – γSub = γPt

Increasing γintIncreasing γsub

Increasing γPt

Modified Wulff Construction (twins)

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

Documents

Wulff Construction