Voronoi construction

With some appropriate definition of nearest neighbors (for example, the Voronoi construction, described in Section III.D, provides a unique definition of nearest neighbors), a bond angle field d can be defined, where 6 is the orientation of a bond between nearest neighbors with respect to some reference direction. In the continuum limit, the bond angle field is related to the displacement field by... 

Our techniques for probing local geometrical and topological structure in the liquid and solid are largely based on the Voronoi construction [187], which is perhaps the most frequently used method for characteriz-... 

The dual of the Voronoi construction, the so-called Delaunay construc-... 

Figurc 18. Voronoi construction for two nearly square arrangements of points, showing how a small displacement of the points can create disclination quadrupole (right) from a configuration with no disclinations (left).

In a cellular model of a glass or liquid derived from the Voronoi construction, each cell i differs in shape, size, and location within it of the molecule. We suppose, following Kirkwood, that each cell i contributes an additive term/, the cellular or local free energy, to the total free energy. We suppose further that the cellular free energy / depends only on the cell volume c, / =/(c,). The total free energy F of (3.1) then becomes... 

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...

A 3-dimensional Voronoi construction was very recently used to model the diffusion of small molecules in polymeric glasses [713]. It will not be surprising if this modeling method will soon be widely accepted. 

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. 

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