Anchored boundary conditions for locally isostatic networks

Finite pieces of locally isostatic networks have a large number of floppy modes because of missing constraints at the surface. Here we show that by imposing suitable boundary conditions at the surface the network can be rendered effectively isostatic. We refer to these as anchored boundary conditions. An important example is formed by a two-dimensional network of corner sharing triangles, which is the focus of this paper. Another way of rendering such networks isostatic is by adding an external wire along which all unpinned vertices can slide (sliding boundary conditions). This approach also allows for the incorporation of boundaries associated with internal holes and complex sample geometries, which are illustrated with examples. The recent synthesis of bilayers of vitreous silica has provided impetus for this work. Experimental results from the imaging of finite pieces at the atomic level need such boundary conditions, if the observed structure is to be computer refined so that the interior atoms have the perception of being in an infinite isostatic environment.

Figure 1

A piece of bilayer of vitreous silica derived from an SPM image [5]. The Si atoms are shown as red discs (in the centers of triangles) and the O atoms (corners of triangles) as black discs. The local covalent bonding leads to the yellow almost-equilateral triangles that are freely jointed, which we will refer to as pinned. The triangles at the surface have either one or two vertices unpinned.

Figure 2

Sliding boundary conditions, used for a piece of the sample shown in Fig. 1. The boundary sites are shown as blue discs and the three purple triangles at the lower right in Fig. 3 have been removed. The red Si atoms at the centers of the triangles in Fig. 1 have also been removed for clarity. The boundary is formed as a smooth analytic curve by using a Fourier series with 16 sine and 16 cosine terms to match the number of surface vertices, where the center for the radius $r\left(\theta \right)$ is placed at the centroid of the 32 boundary vertices [11]. Note that sliding boundary conditions do not require an even number of boundary sites.

Figure 3

The anchored boundary conditions used for the sample shown in Fig. 1. The alternating anchored sites on the boundary are shown as blue discs and the three purple triangles at the lower right are removed to give an even number of unpinned surface sites. The red Si atoms at the centers of the triangles in Fig. 1 have been suppressed for clarity.

Figure 4

The triangle ring network, complementary to that in Fig. 3, where the Si atoms, shown as red discs, at the center of each triangle are emphasized in this three-coordinated network. Dashed edges are shown connecting to the anchored sites.

Figure 5

A typical subgraph from Fig. 3 used in the proof that there are no rigid subgraphs larger than a single triangle. (See Lemma 1.)

Figure 6

Two additional boundary conditions used for the sample shown in Fig. 3, with the three purple triangles at the lower left removed to give an even number of unpinned surface sites. On the left, alternating surface sites are connected to one another through triangulation of first and second neighbors, with the last three connections not needed (these would lead to redundancy). Hence there are three additional macroscopic motions when compared to Fig. 3 which can be considered as being pinned to the page rather than to the internal frame shown by blue straight lines. On the right we illustrate anchoring with additional bars which connect all unpinned surface sites, except again three are absent, to avoid redundancy, and to give the three additional macroscopic motions when compared to Fig. 3.

Figure 7

Two, at first sight, more complex anchored boundary conditions that by our results can be used for the sample shown in Fig. 2, with the three purple triangles at the lower left removed to give an even number of unpinned surface sites. The anchored sites are shown as blue discs, with an even number of surface sites in both graphs. The graph at the right has an even number of surface sites in both the outer and inner boundary. The red Si atoms at the centers of the triangles have been suppressed for clarity. The green line goes through the boundary triangles.

Figure 8

Even more complex boundaries, developed from the sample shown in Fig. 3 by removing triangles to form two internal holes. The boundary sites are shown as blue discs and the three purple triangles at the lower left in Fig. 3 have been removed. The red Si atoms at the centers of the triangles in Fig. 1 have also been removed for clarity. The green line forms a continuous boundary which goes through all the surface sites which must be an even number. The anchored (blue) sites then alternate with the unpinned sites on the green boundary curve which has to cross the bulk sample in two places to reach the two internal holes. Here there are 32 boundary sites, 5 boundary sites in the upper hole and 7 in the lower hole, giving a total even number of 44 boundary sites. Where these crossings take place is arbitrary, but it is important that the anchored and unpinned surface sites alternate along whatever (green) boundary line is drawn.