Void Analysis

Given an interest in voids in aggregated emulsions, it’s important to investigate ways of measuring, counting and otherwise characterizing void structure so that we can measure changes in void structure over time. Void analysis is one of the many avenues that open up once you have the means of extracting the coordinates of the features in the system.

We use two rather different methods to study void structure, the relatively conventional Voronoi analysis, and a novel “kissing spheres” approach based on the generalized Apollonius problem.

For a set of points in a plane the Voronoi decomposition, illustrated to the right, divides the plane into regions such each region encompasses a single point. A region containing a particular point is made up of everywhere in the plane that is closer to that point than to any other. Roughly, this amounts to constructing lines joining all the points to their nearest neighbours (technically, what you need is a Delaunay triangulation) then connecting the mid-points of these lines around a particular point in the plane to give its Voronoi cell.

In the diagram the region outlined in red is the single Voronoi cell corresponding to the point in the middle of the diagram.

The Voronoi decomposition can also be performed in three dimensions, in which case the space is divided up into a set of polyhedra having various numbers of faces, edges and surface areas. These polyhedra fit together exactly to fill the sample space. The set of polyhedra is then a way of characterising the set of points, where for us points means droplet centres.

The “kissing spheres” approach has a lineage dating back to ancient times. The problem as set out in the writings of the Greek geometer Apollonius (Apollonius of Perga, c255-170 BC) is to construct a circle tangent to three given circles in the plane. The generalised (three dimensional) problem that we have developed is to construct a sphere that is tangent to four given spheres. The constructed sphere is described as osculating, meaning to touch without crossing so as to possess a common tangent. Since to osculate is to kiss, this construction is sometimes known as “kissing spheres”.

Kissing spheres have featured in previous IFR research in areas as diverse as artificial lobster reefs to the morphology of granular systems, and was tackled as a computationally expensive set of quadratic simultaneous equations. We have recently extended the analysis to give a solution in terms of a single equation which, though still computationally intensive for the problem in hand, is a great deal more efficient than the previous method. In the present context, the Apollonian sphere construction yields all possible spheres that can fit in the spaces between actual droplets.

Many of the osculating spheres overlap, corresponding to “overlapping holes”. The complete set of overlapping holes we termed “O-holes”. The union of all the O-holes is a way of mapping the complete droplet-free region of the sample. In addition we have “I-holes”, or independent holes, which provide a clearer representation of the void structure. To extract the I-holes, the O-holes are first ranked according to size, and then all those that intersect a larger hole are systematically removed. Single large I-holes will therefore remain in large droplet-free regions, so I-holes give a clear picture of the both the size and location of those parts of the sample that are free of droplets, as the graphic below illustrates.

We prefer the kissing sphere analysis. Though it is computationally more expensive than the Voronoi approach, and therefore takes a significant amount of computer time, the results are more intuitive and easier to interpret. This is even more so for polydisperse systems, since the Voronoi approach only makes use of droplet centre coordinates, whereas the kissing sphere route knows about the entire space occupied by droplets and so is sensitive to the droplet sizes.

The images below show a kissing sphere analysis of a portion of a 30% aggregated oil-in-water emulsion based on explicit droplet locations.

To begin with, here is a reconstruction of the emulsion, with droplets drawn at locations determined by the droplet location algorithm:	Using the kissing spheres approach, the space occupied by the continuous phase can be filled with overlapping holes, shown here in green:
For clarity, wipe out the original droplets themselves to leave only a representation of the continuous phase:	Finally, refine the analysis to represent the continuous phase in terms of non-overlapping holes (green) and completely isolated holes (black)

This construction provides a means of characterizing and charting the morphology of the aggregated network either over time or in response to changes in system parameters.