# Nano- and Micropatterning on Optical Fibers by Bottom-Up Approach: The Importance of Being Ordered
## 4 Methods for Order Quantification
### Order Quantification by Voronoi Tessellation  
A simple and intuitive method for evaluating the degree of order in 2D periodic structures is the one based on Voronoi tessellation. A Voronoi polygon is the smallest convex polygon that surrounds a point so that its sides are perpendicularly bisecting the lines connecting the central point to the ones immediately neighbors. Because every point belonging to a polygon is closer to the center point of that polygon than to every other center points, this technique is commonly used in geographical or networking applications, for example, to locate the closest airport to one position on a map or the closest cell tower to which a mobile phone will connect during a call. On a far more reduced scale, on the other hand, Voronoi tessellation can be used to identify lattice defects. Because Voronoi polygons have a number of sides equal to the number of nearest neighbors of the point in the center, they allow the quantification of the deviation from the expected geometry. This can be intuitively understood by the observation of the diagrams in Figure 6(d-i), showing a Voronoi analysis applied to 1 \(\upmu\)m \(\times\) 1 \(\upmu\)m FE-SEM images of a BCP thin film prepared by spin-coating [100]. If a periodic structure is expected, hexagonal arrangement, in this case, a perfect honeycomb tiling should be observed, with each cell consisting of a regular hexagon. In the example, each different polygon is filled by a different color, and only 35% of them are hexagons, indicated by light green. For a square arrangement, on the other hand, the Voronoi diagram consists of square cells, here filled by dark blue.  
When studying the order of quasi-periodic structures, any deviation from the ideal tessellation can be interpreted as degradation of order. As such, these tools can be used for order quantification. Usually, in Voronoi analysis, order quantification is conducted in terms of conformation entropy (\(S\)) of the pattern [101], where the term entropy is perhaps improperly used to indicate the degree of disorder. It is calculated as  
\[S=-\ \Sigma\ P_{n}lnP_{n} \tag{2}\]  
where \(n\) is the number of sides of the Voronoi polygon (the coordination number), and \(P_{n}\), the fraction of polygons with \(n\). The conformational entropy of the quasi-periodic structure is then compared with the reference value \(S=1.71\), obtained in the case of a totally random distribution of points, and with the null value, ideally obtainable with an array of infinite dimensions and perfectly ordered. As an example of this approach, Figure 6(d-ii) reports SEM images and corresponding Voronoi tessellation of microporous film obtained by breath figure templating. The aim of the authors was to compare different processing conditions to obtain cellulose acetate butyrate (CAB) microporous films: from chloroform solution (top), from THF solution (middle), and from THF by spin-coating (bottom). As in the majority of the self-assembly methods, the expected geometry for breath figure films is the hexagonal one. By using chloroform as a solvent, the probability of having five, six, or seven prime neighbors is respectively \(P5=0.061\), \(P6=0.893\) and \(P7=0.046\), which gives conformational entropy \(S=0.41\), far from the value 1.71 calculated for a completely disordered distribution. For the THF case, \(S\) values are slightly increased to 0.48, while for the breath figure film obtained by spin-coating a THF solution, \(S\) becomes 0.86, indicating that more defects in hexagonal packing are generated by these conditions. This trend is immediately evidenced by the increasing number of colored polygons, indicating packing defects.