Or, “How to make atoms believe they are part of something bigger”.
Over the last year I participated in an interdisciplinary research project with some mathematicians and some materials scientists who study zeolites. We recently published a paper on the topic, which is available here and was published in Physical Reviews E. Below I summarize our key findings. In a way the most interesting part of the project for me was witnessing some of the difficulties in communication that arose over the course of our research. Since we were using tools from discrete geometry to study the atomic structure of the material, it was essential that we were able to formalize the properties of the material in the language of graph theory. It proved challenging to do this in a way that would remain understandable to a wide audience in the physical sciences. Sometimes our “discussions” sounded a little like this:
Materials Scientist: “NO THIS IS WHAT I’M TALKING ABOUT” [pointing to a picture].
Mathematician: “BUT THIS IS WHAT THAT IS!” [pointing to an enumerated list of graph properties].
In the end I think everyone was happy with the outcome, which is a paper containing both pictures and some good definitions, theorems and proofs. A version of the following write-up is to appear in the Notes of the Canadian Mathematical Society.
Boundary conditions are an essential consideration throughout many areas of scientific computer simulation. When modelling materials at the atomic level, we may wish to draw conclusions about properties of the bulk from an observation of a sample. Typically the boundary alters the properties of the sample by the ratio of the number of surface atoms to the number in the bulk. In the thermodynamic limit, as the size of the system increases, this ratio goes to zero. Similar statements are true about the mechanical and vibrational properties of a system, except for those systems which lie on the border of mechanical instability (isostatic systems). In such cases, the boundary conditions are important no matter how many atoms the system contains.
An example of such a system is vitreous silica (SiO2), which can be imagined as two layers of tetrahedra which are mirror images of each other and meet at the apices of each pair of mirrored tetrahedra. Here is an image showing part of a Scanning Probe Microscope image of such a sample (from this source):
We represent the Si atoms as red discs and the O atoms as black discs. Local covalent bonding yields the almost-equilateral triangles, which are freely jointed. The surface triangles have either one or two vertices unpinned, and all other vertices are called pinned.
The fragment of the corner-sharing triangle network shown above has the property that the number of degrees of freedom of the equilateral triangles in the plane is exactly balanced by the shared pinning constraints at each vertex. We call such a network locally isostatic. This condition is destroyed at the boundary, where the unpinned triangles are free to move, and may propagate flexibility toward the inside of the sample. Intuitively, the degrees of freedom of systems like the one shown above correspond exactly to the unpinned triangles, and therefore finding a “good” way to pin these boundary triangles should generate the desired boundary conditions. A rigorous proof of this fact involves showing that there are no additional degrees of freedom resulting from sub-networks which contain more constraints than degrees of freedom.
In our paper we define boundary conditions for graphs which capture the combinatorics of silica bilayers. We then describe two methods to completely immobilize a finite piece of such a network: either by attaching the boundary to a wire rigidly attached to the plane, allowing the boundary vertices to slide along the wire (Figure 3); or by completely immobilizing alternate vertices along the boundary (Figure 4). Both methods apply even when the boundary is quite complex, possibly involving internal holes or missing areas in the structure.
The methods used in our paper come from combinatorial rigidity, which is concerned with the rigidity or flexibility of structures. One of the key ideas is that generically the mechanical properties of an embedded graph is determined by its combinatorics alone. A generic realization is, roughly speaking, one without any special geometry (for instance, no three vertices lie on a line, no four vertices lie on a plane, etc.). The most well-known result in this area is Laman’s theorem, which provides a complete combinatorial characterization of the generic rigidity of graphs in dimension 2.
To describe appropriate boundary conditions for a network like that in Figure 1, we must first define a combinatorial model to describe such a network. We capture this with the graph shown in Figure 2, where the triangles have been replaced with vertices, and an edge connects two vertices if the corresponding triangles are pinned together. We define a triangle ring network to be a graph G such that
1) G has vertices of degree 2 and 3 only, and G is 2-connected,
2) there is a simple cycle C in G that contains all the degree 2 vertices, and there are at least 3 of these,
3) any edge cut set in G that disconnects a subgraph containing only degree 3 vertices has size at least 3.
We now describe two ways of pinning, that is, completely immobilizing the network. A slider constrains the motion of a point to remain on a fixed line, that is rigidly attached to the plane. A network with generically placed sliders is pinned-isostatic if it is pinned, but removing any pin or slider destroys this stability (Streinu-Theran, 2010).
Theorem 1: Adding one slider to each degree 2 boundary vertex of a triangle ring network G gives a pinned-isostatic network (Figure 3).
We next consider completely pinning some of the boundary vertices, which we model combinatorially by placing two sliders at that vertex.
Theorem 2: Let G be a triangle ring network with an even number of degree 2 vertices on the boundary cycle C. Then, pinning every other boundary vertex that is encountered while following C in cyclic order produces a pinned-isostatic network (Figure 4).
The boundary conditions described in Theorems 1 and 2 should be useful in numerical simulations involving finite pieces of locally isostatic networks. Although it is intuitive and natural to think of G as a planar graph with C as the outer boundary, in fact the combinatorial set-up is more general. Figure 4 illustrates a more complex sample with interior holes.