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L21: Topology of

Microstructure

A. D. Rollett

27-750

Spring 2006CarnegieMellon

MRSEC

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Outline

• Objectives• Motivation• Quantities,

– definitions – measurable – Derivable

• Problems that use Topology

• Topology

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Objectives• To describe the important features of grain boundary

networks, i.e. their topology.• To illustrate the principles used in extracting grain

boundary properties (e.g. energy) from geometry+crystallography of grain boundaries: microstructural analysis.

• To understand how Herring’s equations lead to a method of obtaining (relative) grain boundary (and surface) energies as a function of boundary type.

• To understand how curvature-driven grain boundary migration leads to a method of obtaining (relative) mobilities.

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

4 Topology - why study it?

• The behavior of networks of interfaces is largely driven by their topology. The connectivity of the interfaces matters more than dimensions.

• Example: in a 2D boundary network, whether a grain shrinks or grows depends on the number of sides (von Neumann-Mullins), not its dimensions (although there is a size-no._of_sides relationship).

• In a 2D grain boundary network, the dihedral angles at triple junctions is 120°. Why? There is a force balance at each triple junction and symmetry dictates that the angles must be equal, therefore each 120°.

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

5 Shrink vs. Grow (Topology)

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

Read: Mullins, W. W. (1956). "Two-dimensional motion of idealized grain boundaries." J. Appl. Phys. 27: 900-904; also Palmer M, Nordberg J, Rajan K, Glicksman M, Fradkov VY. Two-dimensional grain growth in rapidly solidified succinonitrile films. Met. Mater. Trans. 1995;26A:1061.

Triplejunction

6 Topology of Networks in 3D

• Consider the body as a polycrystal in which only the grain boundaries are of interest.

• Each grain is a polyhedron with facets, edges (triple lines) and vertices (corners).

• Typical structure has three facets meeting at an edge (triple line/junction or TJ); Why 3-fold junctions? Because higher order junctions are unstable [to be proved].

• Four edges (TJs) meet at a vertex (corner).

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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Definitions

• G B = Grain = polyhedral object = polyhedron = body

• F = Facet = face = grain boundary• E = Edge = triple line = triple junction = TJ• C V = Corner = Vertex = points• n = number of edges around a facet• overbar or <angle brackets> indicates

average quantity

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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Euler’s equations

• 3D: simple polyhedra (no re-entrant shapes)V + F = E + 2 [G = 1]

• 3D: connected polyhedra (grain networks)V + F = E + G +1

• 2D: connected polygons V + F = E + 1

• Proof: see What is Mathematics? by Courant & Robbins (1956) O.U.P., pp 235-240.

• These relationships apply in all cases with no restrictions on connectivity.

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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Grain Networks• A consequence of the characteristic that

three grain boundaries meet at each edge to form a triple junction is this:

3V = 2E

E

EE

E

EE

EE

E

EEEV

V

V

V

V

V

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

This relationship is particular to grain networks. It is not general because it depends on junctions being triple junctions only (no quadri-junctions, for example).

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2D sections• In a network of 2D grains, each grain boundary

has two vertices at each end, each of which is shared with two other grain boundaries (edges):

2/3 E = V, or, 2E = 3V E =

1.5V• Each grain has an average of 6 boundaries

and each boundary is shared: <n> = 6: E = <n>/2 G =

3G, or, V = 2/3 E = 2/3 3G

= 2G

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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2D sections

6-sided grain= unit cell;each vertex has 1/3 in eachunit cell;each boundaryhas 1/2 in each cell

split each edge

divide each vertex by 3

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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2D Topology: polygons

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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3D Topology: polyhedra

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

14 Application: regular

shapes• For grains in polycrystalline

solids, the shapes are approximated by tetrakaidecahedra: -ttkd to

-ttkd.

(a) soap froth; (b) plant pith cells; (c) grains in Al-Sn

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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3D von Neumann-Mullins

• The existence of the n-6 rule in 2D suggests that there might be a similar rule in 3D.

• The foundation of the n-6 rule is the integration of curvature around the circumference of a grain. In 3D it turns out that integrating the mean curvature around a polyhedron is not possible mathematically. Various approximations have been tried (see, e.g., Hilgenfeldt, S., A. Kraynik, S. Koehler and H. Stone (2001). "An accurate von Neumann's law for three-dimensional foams." Physical Review Letters 86(12): 2685-2688.).

• Empirically, however, it turns out that there is an F-13.6 rule that holds, based on several computer simulations. Grains with 14 or more facets grow (on average) whereas grains with 13 or fewer facets shrink.

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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3D von Neumann-Mullins

• Glicksman recently developed a theory based on polyhedra with regular shapes. This predicts a crossing point at N=13.4.

See: Glicksman ME, “Analysis of 3-D network structures “;PHILOSOPHICAL MAGAZINE 85 (1): 3-31 JAN 1 2005.

Also, Glazier JA. “Grain growth in three dimensions depends on grain topology.” Phys. Rev. Lett. 1993;70:2170.

17 3D vs 2D: polygonal

faces• Average no. of

edges on polygonal faces is less than 6 for typical 3D grains/cells.

• Typical <n>=5.14• In 2D, <n>=6.

Objectives Notation Equations Delesse SV-PL LA-PL Topology Grain_Size Distributions

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Summary

• Grain boundary networks obey certain geometrical rules. For example, in 2D, grains have an average of exactly 6 sides.

• Boundaries generally meet at triple lines, which simplifies some of the topological relationships.

• Nearly all cellular materials (including biological ones) have very similar topologies.

• Force balance (Herring relations) at Triple Junctions determines dihedral angles (120° for isotropic boundaries), which in turn determine boundary curvatures for moving boundaries.