Characterisation of Disordered Structures

In this thesis I will look at how large, complex structures can be interpreted and evaluated using an information theoretic approach. The work specifically investigates techniques to understand disordered materials. It explains a novel framework using statistical methods to investigate structural information of very large data sets. This framework facilitates understanding of complex structures through the quantification of information and disorder. Large scale structures including granular media and amorphous atomic systems can also be processed. The need to deal with larger complex structures has been driven by new methods used to characterise amorphous materials, such as atomic scale tomography. In addition, computers are allowing for the creation of larger and larger data sets for researchers to analyse, requiring new techniques for storing and understanding information.

As it has become possible to analyse large complex systems there has been a corresponding increase in attempts to scientifically understand these systems. New, man-made, complex systems have emerged such as the stock market and on-line networks. This has boosted interest in their interpretation, with the hopes they can be more easily manipulated or controlled. Crystallography has been applied to great effect in biology, having been used to discover the structure of DNA and develop new drugs (UNESCO,2013). However it only describes crystal structure, which can be a drawback as a large majority of matter is amorphous. As such it is hoped that interpreting and understanding disorder may lead to similar breakthroughs in disordered materials.

Entropic measures such as the mutual information and Kullback Leibler Divergence are used to investigate the nature of structural information and its impact on the system. I examine how this information propagates in a system, and how it could quantify the amount of organisation in a system that is structurally disordered. The methodology introduced in this thesis extracts useful information from large data sets to allow for a quantification of disorder. The calculated entropy for amorphous packings is generally less than 1 bit with Mutual information between 0 and 0.1 bits. The results verify direct correlation between Mutual Information and the correlation coefficient using various techniques. The Mutual information shows most information is obtained where sphere density is highest, following a similar trend to that of the Radial distribution function, and generally increasing for higher packing fractions. Evidence of the Random Close Packed (RCP) and Random Loose Packed (RLP) limits in two dimensions is shown, as well as evidence of both phases in time-lapsed 3D packings.

The Kullback Leibler Divergence is also explored as a relative measure of disorder. This is achieved by calculating redundant information in packings so that areas of low and high order can be shown. Results present colour maps displaying relative information in random disk packings from which motifs can be identified. For higher packing fractions distinct borders form for areas of low and high information, particularly where crystallisation has occurred. Again, these results show an increase in information for more densely packed structures, as expected, with a Kullback Leibler divergence of between 0 and 1 bits.

Finally I introduce the concept of self-referential order which provides a way to quantify structural organisation in non-crystalline materials, by referencing part of the system in a similar way to a unit cell. This allows a step forward in understanding and characterising disorder, helping to develop a framework to encode amorphous structures in an efficient way. These results show increasing information for higher packing fractions as well as further evidence of RLP and RCP limits around packing fractions of 0.54 and 0.64 respectively.