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Skeletons in the Labyrinth Skeletons in the labyrinth Medial Representations and packing properties of bicontinuous space partitions by Gerd Elmar Schr¨oder A thesis submitted for the degree of Doctor of Philosophy at The Australian National University Canberra, 18 February 2005 (submission date) © Gerd Elmar Schr¨oder A Typeset in Palatino by TEXand LTEX 2 . Except where otherwise indicated, this thesis is my own original work. Gerd Elmar Schr¨oder 18 February 2005 Acknowledgments Many people have helped and inspired me through my doctoral studies. There are many who should be thanked for all the words of encouragement and interest in my work. At the risk of leaving somebody out, I would like to particularly acknowledge the following friends and institutions: My foremost thanks and acknowledgment is to my principal supervisor Stephen Hyde who has been a most helpful and relaxed mentor. I felt inspired when I first talked to him a few years ago, and continue to do so. He has also become an alright friend. I am grateful to my advisors Adrian Sheppard, Tomaso Aste, Stuart Ramsden and Andrew Christy for much help and for countless suggestions that I have incorporated into my work. Stuart has taught me the art of visualisation, and Andrew the mysteries of space groups. I have greatly benefitted from discussions with Klaus Mecke, in particular from his critical comments on the sensitivities of the medial surface. I much regret that I have not managed to make our joint projects part of this thesis, and am much looking for- ward to a prolonged collaboration. I thank Akasaka Satoshi and Hasegawa Hirokazu (Kyoto University) for providing me with the triblock copolymer dataset, and for helpful discussions during its analy- sis. I acknowledge financial support in form of a travel scholarship from the National In- stitute of Physical Science for a visit to the Mathematical Science Research Institute (California). During that visit I enjoyed fruitful conversations with Nina Amenta. I am grateful to the Max-Planck Institut for Complex Systems in Dresden for generous financial support to participate in the workshop “Geometry and Mechanics of struc- tured materials” in September 2002. I thank Nick Rivier for the interest he showed in my work during that workshop, and the suggestions he made. I gratefully acknowledge the support of AusIndustry and the German Academic ex- change service (DAAD) through a German-Australian grant scheme that has made possible a number of visits to Klaus Mecke at Siegfried Dietrich’s group at the Max- Planck Institut in Stuttgart. I acknowledge financial support of the Australian supercomputer facility to partici- pate in the Australian Partnership for Advanced computing conference at the Gold Coast in October 2003. I am grateful to the Australian government for an International Postgraduate Research Scholarship (IPRS) to cover my tuition and to the Australian National University for an ANU PhD scholarship covering my living allowance. Vanessa Robins, Mohammad Saadatfar and Judit O’Vari have been of much help in proof-reading the manuscript. The Department of Applied Mathematics has been a fantastic place to work and study, and all of its members wonderful people to have around. The interdisciplinarity and level of communication is amazing, the friendship, interest and care towards students is humbling. I feel I have learned a lot, while having a great time. My parents, Bernd and Hannelore, deserve my deepest gratitude for their generous financial supportand I am indebtedto them, my brother and my sisters, and my grand mother for doing without me, so far away for so long. I owe a very special thank-you to Cate Turk, now my wife, who has helped and sup- ported me greatly throughout the second half of my thesis research, in the best of all sorts of ways. Without her assistance in the final week before submission, this thesis would have been a great deal more disordered, and I might have lost my sanity. She has also suggested the title. Abstract This thesis discusses medial skeleton representations of labyrinthine domains. These generalised channel-graph representations succinctly characterise both topology and geometry of bicontinuous space partitions and provide quantitative measures of ex- trinsic packing properties beyond volume-to-surface ratios. Analyses are presented of infinite periodic minimal surfaces as model structures for bicontinuous liquid crys- talline mesophases and of electron-tomography data of a novel triblock copolymer network phase. We describe the construction of both a geometrically-centered 1D line graph and a 2D medial surface skeleton of 3D labyrinthine domains. The medial surface is a branched surface structure centered within the domain, resulting from collapse of parallel sur- faces of the domain’s bounding surface Ë . Together with an associated distance func- tion it affords a complete geometric description, that means the domain can be re- constructed from it as a union of balls centered on the MS. The radius of these balls provides a concise definition of local channel thickness. We use the variations of the ball radii Ð to quantify packing homogeneity (as opposed to ¿ curvature homogeneity) of labyrinthine domains in E . We define the degree of pack- Ë ing homogeneity as the width of the distribution of Ð over , with perfect homogeneity Ð corresponding to constant Ð . The average can be interpreted as the average channel Ð ´Î µ diameter, allowing for the definition of an average shape parameter . These concepts are applicable to self-assembly processes of liquid crystalline mesophases from nearly mono-disperse building blocks of typical molecular dimensions. Analyses are presented for families of infinite periodic minimal surfaces. Among the cubic Primitive, Diamond and Gyroid surfaces and their tetragonal and rhombohe- dral distortions, the cubic Gyroid is shown to be the most homogeneous, for various physically motivated normalisations of the length scale. We describe Voronoi-based algorithms for MS computations of triangulated surface data, including a novel adaption particularly suitable for exact data from mathemati- cal parameterisations. An algorithm to obtain evenly triangulated representations of surfaces is also described. The medial surface construction is also successfully applied to the interpretation of electron-tomography data of a novel symmetric but non-cubic network phase (or- viii thorhombic symmetry ) in a linear triblock copolymer melt. This analysis demon- strates that the medial surface is a useful tool for spatial structure recognition of space- filling hyperbolic structures as it combines complete geometric and topological de- scription with the spatial sparsity of its representation (as compared to the bounding surface). This holds true despite its sensitivity to noise. The network phase identified in a PS-block-PI-block-PDMS copolymer blend consists of two identical intertwined networks of eight-rings with three- and four-coordinated nodes (”tfa” in O’Keeffe’s notation) with likely symmetry Fddd and c/a ratio of about 4. It is different from another Fddd phase (based on a single three-coordinated net- work) recently proposed for a similar system [54]. Date of this version: 19 September 2006, with small changes compared to the accepted version of this thesis Contents Acknowledgements v Abstract vii 1 Introduction 1 1.1 FrustrationinSelf-assembledmesophases . ....... 4 1.2 MSconcepttocharacterisechannelgeometries . ...... 6 1.3 MS-likestructuresandlineskeletonsinphysics . ....... 10 Organisationofthisthesis . 12 2 Medial Surfaces 15 2.1 DefinitionandbasicpropertiesoftheMS . 15 2.2 SomeresultsandliteratureabouttheMS . 19 2.3 Voronoidiagramsandalgorithms. 21 2.4 Voronoibasedmedialsurfacealgorithms . ..... 25 2.5 MSoflabyrinthinestructures . 27 2.6 Voronoi-basedMSforexactsurfacedata . 29 2.6.1 Analysisofthenumericalrobustness . 33 2.7 OtherMSalgorithmsanddomainrepresentations . ..... 37 3 Line graphs on the medial surface 39 3.1 RelatedWork .................................. 41 3.2 Topological equivalence of the domain and its line graph ......... 43 3.3 Euclidean distance maps and geometrically centered curves ....... 44 3.4 Relation between saddle points of and ................. 46 3.5 Formal definition of a geometrically centered line graph . ........ 53 x Contents 3.6 Computationofthelinegraph . 57 3.6.1 Detection of saddle points of the distance function on Ë ...... 58 ¿ E 3.6.2 Saddle points of the Euclidean distance map in ....... 63 3.6.3 Detectionoflinesofsteepestascent . 63 3.7 Representabilityofalabyrinthbyalinegraph . ....... 65 3.8 Images of line graph edges on Ë andwatershedpartitions . 67 4 Representations of Infinite Periodic Minimal Surfaces 71 4.1 BasicpropertiesofIPMS . 71 4.2 HistoryofandliteratureonIPMS. 73 4.3 WeierstrassparametrisationofIPMS . 75 4.4 Numericalintegration . 80 4.5 ParametrisationsofspecificIPMS. 83 4.5.1 The cubic Primitive, Diamond and Gyroid surfaces . 83 4.5.2 TherhombohedralrPDsurfacefamily . 86 4.5.3 TherhombohedralrGsurfacefamily. 88 4.5.4 ThetetragonaltDandtPsurfacefamilies . 89 4.5.5 ThetetragonaldistortiontGoftheGyroid . 90 4.5.6 ThecubicI-WPsurface. 91 4.6 EventriangulationsofIPMS. 92 5 MS Analysis of cubic IPMS 99 5.1 Homogeneityandpackingfrustration . 100 5.2 Relevance to liquid crystalline mesophase formation . ..........106 5.3 RelativelengthscalesofIPMS. 111 5.4 RelativehomogeneityoftheP,DandGsurfaces . 112 5.5 MSgeometryofthecubicP,DandGsurfaces. 115 5.6 MSgeometryoftheI-WPsurface . 122 5.7 Finalthoughts..................................124
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