On the three-dimensional twin-limited microstructure

Appendix: Proof of the contiguity of Σ9 boundaries in a twin-limited microstructure

We have asserted, mirroring previous observations [3], that the set of Σ9 boundaries in a three-dimensional microstructure containing only perfect Σ3 and Σ9 boundaries forms a continuous manifold. We sketch here a proof of this theorem under certain fairly unrestrictive assumptions about the grain boundary topology, along with counterexamples that may be generated when these assumptions are violated. A pair of Σ9 boundaries shall be called contiguous if there is a continuous path of Σ9 boundaries linking them.

The assumptions are as follows. The list has been made slightly redundant in order to improve clarity. First, we continue to make the same assumptions as before, which are described in the beginning of the Analysis Section. We further assume that each grain boundary participates in triple junctions, that the grain boundary network is connected, that each triple junction links exactly two quadruple nodes,that each grain is of finite extent, and that the surface of the polycrystal is far outside the region of interest. Finally, we assume that each Σ3 grain boundary is simply connected, which for our purposes means it is topologically equivalent to a polygon.

These assumptions carry several implications, the first being that the grain boundary network is essentially three-dimensional, i. e., we are not talking about a two-dimensional network of columnar grains. The theorem does not hold in the two-dimensional case, as can easily be shown by example (see, e. g., Figs. 6 and 7 in Ref. [3]). Further, the set of grain boundaries is connected; there are, for example, no single grains spanning the polycrystal, separating the network in two, which would provide a trivial counterexample. The uninteresting case of a single grain entirely contained within another grain is also removed from consideration.

It is also implied that there are no prismatic grains such as in Fig. 7b, since this structure includes a closed-loop triple junction that participates in no quadruple nodes. This deserves some comment, since such grains do appear in real materials. It is simple to produce a counterexample to the theorem if we allow this structure, since one may have the same topology with the basal plane converted to a (non-simply-connected) Σ3 and either of the other grain boundaries (the bottom of the prism, or the sides and top of the prism) converted to an isolated R9. However, in some sense, this prismatic grain does not take part in the grain boundary network in an essential way; it can be removed without any significant alteration of the remaining network. No quadruple nodes would be altered, the connectivity of the remaining triple lines would be unchanged, and the percolation behavior in the rest of the grain boundary network would remain the same. In this and similar cases, the theorem holds not for the entire grain boundary network but instead for the largest subset that satisfies the assumptions of the theorem, and we recognize that there may be partially isolated and non-essential parts of the full network that include counterexamples. A more complex counterexample is shown in Fig. 8. This example illustrates the need to restrict non-simply-connected Σ3 boundaries (restricting loop triple junctions is not sufficient), as it is the cylindrical twin boundary CE that makes the construction possible. This is the simplest of this class of counterexamples, yet is already somewhat contrived and probably not of much practical interest.

Fig. 8

Prismatic (cylindrical) grain C capped with two grains A and B, all placed inside grain E so that only grain C intersects the outer boundary of grain E: (a) 3D schematic and (b) cross-section parallel to grain C axis. Not to overcomplicate the figure, the grain boundaries here are drawn rounded rather than faceted as, e. g., in Fig. 7 above; this does not change the topology of the construction. Orientations A, B, C and E are defined by Eqs. 1 and 3; then, boundaries AE, BE and CE are Σ3’s, while AB, AC, and BC are an isolated group of Σ9’s. The construction in (c) shows how additional grains A′ and B′ (with the same orientations as A and B) may be added outside the central construction to satisfy the conditions of the theorem. Provided this core remains unchanged, additional grains may now be added at will without compromising the construction.

Now that we have explained the context and limitations of the theorem, we proceed to a sketch of the proof itself. Consider the representation of the grain boundary network as a graph in which the nodes are the quadruple points and the lines connecting the nodes are triple junctions. It follows from our assumptions that this graph is not empty, that every triple junction appears in the graph, and that every grain boundary takes part in at least one triple junction (otherwise it would be an entirely isolated boundary).

Every Σ9 grain boundary in a quadruple node is contiguous with every other Σ9 grain boundary in the node (since in a microstructure consisting only of Σ3 and Σ9 boundaries, only two quadruple nodes are allowed, the all- Σ9 and the node shown in Fig. 1, and both have this property). Thus, if two triple junctions, each including a Σ9 boundary, meet in a quadruple node, then those two Σ9 boundaries are contiguous. Since every triple junction in a twin-limited structure contains a Σ9 boundary, and since contiguity is an equivalence relation, it follows that two Σ9 boundaries form part of a contiguous set of Σ9 boundaries if there is a path in this triple junction graph that links them. So, if the graph is connected (i. e., there is a path linking any two nodes), then the set containing every Σ9 boundary in the polycrystal is contiguous.

There are only two ways to make the graph disconnected. One is for the grain boundary network itself to be disconnected. In this case, the proposition is trivially false – if the full set of grain boundaries is not contiguous, and each separate region contains Σ9 boundaries, then clearly the Σ9’s cannot form a contiguous manifold. We have already explicitly removed this case from consideration.

Assuming then that the set of grain boundaries is connected, the only other way for the triple-junction graph to be disconnected is for a grain boundary to have two (or more) disconnected edges. Each edge consists of a cyclic alternating sequence of triple-junction lines and quadruple nodes, which in the triple-junction graph would make it a polygon. So, a way to construct a counterexample is to include grain boundaries that are multiply connected, i. e., they are topologically equivalent not to a polygon but to an open-ended (and possibly branched) tube. Then, via constructions such as the one in Fig. 8, we could effectively separate two parts of the network, provided the multiply connected boundaries linking the two parts are all Σ3’s (for if one is a Σ9, then we have regained contiguity). This class of constructions is of mathematical interest only and is unlikely to have much practical consequence.

In summary, in any truly three-dimensional twin-limited polycrystalline structure, the set of Σ9 boundaries is contiguous in any subset of the network obeying the conditions of the theorem. Several classes of counterexamples may be produced by including networks that do not satisfy the conditions. Probably, the only nontrivial, physically realistic counterexample is the prismatic grain, such as in Fig. 7b (and variants of this type of structure), with the practical implications limited by the fact that such grains do not participate in the larger network in an essential way. All other counterexamples are either two-dimensional, trivial, or contrived.

The corresponding proof for the contiguity of the Σ3 boundaries fails for the simple reason that not all triple junctions must contain a Σ3. On the contrary, Σ 9 –R9 –R9 junctions are geometrically necessary in 3D twin-limited microstructures.