Voronoi networks

Until now, the theoretical discussion has focused on monodisperse two-dimensional model systems. However, some studies have been performed on polydisperse systems, notably by Weaire et al. [68-72] The evolution of a soap froth of random cell sizes and shapes, known as a Voronoi network, was simulated by computer [68] (Fig. 7). The condition that three films must always meet at angles of 120° was again used. Cells with more than six sides were found... 

Vrettos, N.A. Imakoma, H., and Okazaki, M., Transport properties of porous media from the microgeometry of a 3-D Voronoi network, Chem. Eng. Process., 26(3), 237-247 (1989). 

Fio. 17. Computer generated structure of hexagonal and random closed-cell foam obtained by Voronoi tessellation, shown as voxel representation of phase function (left), and network diagram where nodes correspond to cells and bonds to cell walls (from Salejova et ah, 2005). 

These values indicate that spherical objects prefer a body-centred cubic lattice, since this lattice maximises the number of faces in the Voronoi cell. Similarly, where the interface is cylindrical, a (two-dimensional) hexagonal network is expected. These arrays are indeed ttiose found in practice [45]. 

The algorithms of Voronoi polyhedra construction may easily be used for the high-throughput analysis of new materials to identify the presence ofan infinite network of channels with suitable size for ionic conductivity or molecular diffusion. At the same time, the approach has a limited applicability for those systems where the topological and geometric factors do not play a significant role, such as proton conductors, or where the framework is not truly immobile and participates in the conduction process, for example, via a paddle-wheel mechanism. 

Fig. 5.3. A two-dimensional system with its network of Voronoi polygons. Heavy lines boundaries of the Voronoi polygons. Light lines the dual network of polygons, constructed by lines kj connecting pairs of particles which contribute a boundary for the Voronoi polygons of i and j.

Once a pore space is known, a pore network can be created through a variety of methods. Methods that reduce the pore space into a topologically equivalent skeleton involve either a thiiming algorithm or, in the case of the 2D models " -a Voronoi diagram around the material locations. An alternative method to determine the representative pore network for a pore space is the maximal ball method, which is a computationally inexpensive techniqne that has been demonstrated for GDL-like structures. ... 

Using the Voronoi tessellation concept, we have constructed two-dimensional models of some pre-gel FPs and their aggregates, and of the mature gelled network. They are shown in Fig. 32a, b. In Fig. 32 the darkly shaded cells are those containing additional FPs, enmeshed or occluded in the network, but not part of it. The unshaded cells stand for voids created by local depletion of... 

Fig. 32a, b. Two-dimensional Voronoi construction models a pre-gel FPs and aggregates b the final network... 

Fig. 33. Two-dimensional Voronoi constructions of two mature networks. Lefthand side small precursor FPs. Righthand side large precursors FPs...

We believe that the fractal character of the precursor polymers and final gelled networks can well be accommodated within the framework of the Voronoi constructs, especially when the polymers and their networks are rigid. 

Schroder C, Neumayr G, Steinhauser O (2009) On the collective network of ionic liquid/ water mixtures. III. Structural analysis of ionic Uquids on the basis of Voronoi decomposition. J Chem Phys 130(19), 194503. doi 10.1063/1.3127782... 

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