L21: Topology of Microstructure

1. L21: Topology of Microstructure A. D. Rollett 27-750 Spring 2006

2. Outline • Objectives • Motivation • Quantities, • definitions • measurable • Derivable • Problems that use Topology • Topology

3. 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-PLLA-PL Topology Grain_Size Distributions

5. Triplejunction Shrink vs. Grow (Topology) 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. Objectives Notation Equations Delesse SV-PLLA-PL Topology Grain_Size Distributions

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-PLLA-PL Topology Grain_Size Distributions

7. 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-PLLA-PL Topology Grain_Size Distributions

8. 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-PLLA-PL Topology Grain_Size Distributions

9. 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 E V V E E E E V V E 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). E E V V E E E Objectives Notation Equations Delesse SV-PLLA-PL Topology Grain_Size Distributions

10. 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-PLLA-PL Topology Grain_Size Distributions

11. 2D sections split each edge 6-sided grain= unit cell;each vertex has 1/3 in eachunit cell;each boundaryhas 1/2 in each cell divide each vertex by 3 Objectives Notation Equations Delesse SV-PLLA-PL Topology Grain_Size Distributions

12. 2D Topology: polygons Objectives Notation Equations Delesse SV-PLLA-PL Topology Grain_Size Distributions

13. 3D Topology: polyhedra Objectives Notation Equations Delesse SV-PLLA-PL Topology Grain_Size Distributions

14. Application: regular shapes • For grains in polycrystalline solids, the shapes are approximated by tetrakaidecahedra: a-ttkd to b-ttkd. (a) soap froth; (b) plant pith cells; (c) grains in Al-Sn Objectives Notation Equations Delesse SV-PLLA-PL Topology Grain_Size Distributions

15. 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 Letters86(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-PLLA-PL Topology Grain_Size Distributions

16. 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-PLLA-PLTopology Grain_Size Distributions

18. 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.