As already said, Surface Evolver can also calculate approximations to periodic sur-faces like the Kelvin foam and the Weaire-Phelan foam. In this section we will explain briefly how this is done.
4.2.1 Computation of the Kelvin and Weaire-Phelan Foams
The input datafiles for the Kelvin and the Weaire-Phelan foams are included in the sample files that come with Surface Evolver. The files are called twointor.fe (Kelvin foam) and phelanc.fe (Weaire-Phelan foam).
Table 4.1 shows the input file for the Kelvin foam. The input file starts with the keyword TORUS_FILLED2. This indicates that we want to calculate a periodic surface. After the keyword periods the basis vectors of the unit cell are defined.
The unit cell can be an arbitrary parallelepiped. In our case it is simply a cube of dimension 1x1x1.
Next the vertices are defined. The vertices all lie in the unit cell. However, this is not necessary. In Figure 4.3 one of the tetrakaidecahedra we want do define and the cubic unit cell are shown. Moreover, the positions of the 12 vertices defined in
TORUS_FILLED
edges /* with torus wrap symbols */
1 1 2 * * *
Table 4.1: twointor.fe - Surface Evolver input datafile for the Kelvin foam; A few lines have been removed.
4 The Surface Evolver
the input file are marked with small spheres. At first sight one might think that 12 more vertices are missing in Figure 4.3. However, if we shift the unit cell by one periodicity vector in each direction we can see that no more than these 12 vertices are required.
Figure 4.3: Tetrakaidecahedron in the unit cell; The posi-tions of the 12 vertices defined in the input file are marked with small spheres.
Next, 24 edges are defined. As we want to define a periodic surface we need a possibility to say how the surface wraps around the unit cell. This is why Surface Evolver uses “torus wrap symbols” to define how edges cross the faces of the unit cell:
* does not cross face of the unit cell,
+ crosses in same direction as periodicity vector, - crosses in opposite direction as periodicity vector.
For example edge number 6 runs from vertex 2 to vertex 7. It does not cross a face in x-direction, it does cross a face in negative y-direction and it does not cross a face in z-direction.
Next 14 faces are defined as a loop over the corresponding edges. Finally two bodies(two tetrakaidecahedra) are defined by listing the boundary faces.
Why two bodies?
It is not sufficient to define one tetrakaidecahedron as one tetrakaidecahedron can-not fill the cubic unit cell. If we look at Figure 4.3 we can see that one tetrakaidec-ahedron is sitting in the middle, and one eighth of a tetrakaidectetrakaidec-ahedron is needed for filling up the eight corners of the cubic unit cell.
The lines defining the two bodies in the input file end with volume 0.500. This means that the volumes of both bodies are constrained to be 0.5. When Surface Evolver tries to minimize the energy it must keep the volumes at 0.5. (The volume of the cubic unit cell is 1. As two tetrakaidecahedra fill one unit cell, the volume
Computation of the Weaire-Phelan foam:
The structure of the input file for the Weaire-Phelan foam (phelanc.fe) is the same as described above. Naturally, it is much longer. 46 vertices, 92 edges, 54 faces and 8 bodies are used. The bodies defined in the input file have been generated by a Voronoi tessellation and do not have exactly equal volumes. However, Surface Evolver will relax the structure and equalize the volumes of the cells at the same time. Various pictures of the resulting Kelvin and Weaire-Phelan foams have been shown in the previous chapter.
4.2.2 Automatic Generation of Surface Evolver Input Files
It is obvious that generating an input file like the one listed in Table 4.1 will be very difficult to do by hand. Therefore, J. M. Sullivan has written a program called vcs which computes three-dimensional Voronoi tessellations and can format the output as a Surface Evolver input file. All that vcs needs for generating the input file for the Weaire-Phelan foam are the coordinates of the eight Voronoi sites (see Table 3.1).
vcs is freely available3. However, the program’s C code is from 1988. We tried to compile the code, but were not successful. The code would require some updates, so that it can be compiled with a current C compiler.
4.2.3 Reported Areas and the Isoperimetric Quotient
The surface area that Surface Evolver reports after each iteration is the area of the surface within one unit cell. (We will use this number later to calculate the volume of the bulk material by multiplying the surface area within one unit cell with the thickness of the shell elements.) After evolving one of the foams it is a good idea to compare the surface area reported by Surface Evolver with values given in the literature e.g. [Weaire & Phelan, 1994].
In the literature normally no areas are reported, but a figure of merit, the so-called isoperimetric quotient is used. The isoperimetric quotient is defined as:
I = 36πV2
A¯3 (4.1)
In (4.1) ¯Ais theaverage cell surface areaandV is the cell volume. The isoperimetric quotient is defined in such a way that it is 1 for a sphere. (Naturally, spheres are non-space-filling.) The higher the isoperimetric quotient the lower is the surface area (or energy) of the structure.
3http://torus.math.uiuc.edu/jms/software/
4 The Surface Evolver
Every facet in Surface Evolver is shared by two bodies, but Surface Evolver counts the area of the facet only once. So if we want to calculate the average cell surface area we have to multiply the areaASE reported by Surface Evolver by 2 and divide by the number of cells ncells in the unit cell:
A¯= 2ASE
ncells
(4.2) The cell volume is defined as a volume constraint in the input file, and it is of course the volume of the unit cell divided by the number of cells in the unit cell.
Using the areas reported by Surface Evolver and the formulas above we get:
IKelvin ≈0.757 (4.3)
IWeaire-Phelan ≈0.764 (4.4)
These values are also found in the literature e.g. [Weaire & Phelan, 1994].