1 L21: Topology of Microstructure A. D. Rollett 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 S V -P L L A -P L 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 S V -P L L A -P L Topology Grain_Size Distributions

5 Shrink vs. Grow (Topology) Objectives Notation Equations Delesse S V -P L L A -P L Topology Grain_Size Distributions Read: Mullins, W. W. (1956). "Two-dimensional motion of idealized grain boundaries." J. Appl. Phys. 27: ; 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. Triple junction

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 S V -P L L A -P L 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 indicates average quantity Objectives Notation Equations Delesse S V -P L L A -P L 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 These relationships apply in all cases with no restrictions on connectivity. Objectives Notation Equations Delesse S V -P L L A -P L 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 E E E E E E E E E E V V V V V V Objectives Notation Equations Delesse S V -P L L A -P L 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).

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: = 6: E = /2 G = 3G, or, V = 2/3 E = 2/3 3G = 2G Objectives Notation Equations Delesse S V -P L L A -P L Topology Grain_Size Distributions

11 2D sections 6-sided grain = unit cell; each vertex has 1/3 in each unit cell; each boundary has 1/2 in each cell split each edge divide each vertex by 3 Objectives Notation Equations Delesse S V -P L L A -P L Topology Grain_Size Distributions

12 2D Topology: polygons Objectives Notation Equations Delesse S V -P L L A -P L Topology Grain_Size Distributions

13 3D Topology: polyhedra Objectives Notation Equations Delesse S V -P L L A -P L 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 S V -P L L A -P L 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 Letters 86(12): ). Empirically, however, it turns out that there is an F 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 S V -P L L A -P L 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 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 =5.14 In 2D, =6. Objectives Notation Equations Delesse S V -P L L A -P L Topology 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.