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Problems at the Nexus of Geometry and Soft Matter: Rings, Ribbons and Shells

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Problems at the Nexus of Geometry and Soft Matter: Rings, Ribbons and Shells Problems at the Nexus of Geometry and Soft Matter: Rings, Ribbons and Shells The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Yong, Ee Hou. 2012. Problems at the Nexus of Geometry and Soft Matter: Rings, Ribbons and Shells. Doctoral dissertation, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:9414558 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA c 2012 - Ee Hou Yong All rights reserved. Thesis advisor Author L. Mahadevan Ee Hou Yong Problems at the Nexus of Geometry and Soft Matter: Rings, Ribbons and Shells Abstract There has been an increasing appreciation of the role in which elasticity plays in soft matter. The understanding of many shapes and conformations of complex sys- tems during equilibrium or non-equilibrium processes, ranging from the macroscopic to the microscopic, can be explained to a large extend by the theory of elasticity. We are motivated by older studies on how topology and shape couple in di↵erent novel systems and in this thesis, we present novel systems and tools for gaining fundamental insights into the wonderful world of geometry and soft matter. We first look at how defects, topology and geometry come together in the physics of thin membranes. Topological constraint plays a fundamental role on the morphology of crumpling membranes of genus zero and suggest how di↵erent fundamental shapes, such as platonic solids, can arise through a crumpling process. We present a way of classifying disclinations using a generalized “Casper-Klug” coordination number. We show that there exist symmetry breaking during the crumpling process, which can be described using Landau theory and that thin membranes preserve the memory of their defects. Next we consider the problem of the shapes of Bacillus spores and show how one can understand the folding patterns seen in bacterial coats by looking at the simplified iii Abstract iv problem of two concentric rings connected via springs. We show that when the two rings loses contact, rucks spontaneous formed leading to the complex folding patterns. We also develop a simple system of an extensible elastic on a spring support to study bifurcation in system that has adhesion. We explain the bifurcation diagram and show how it di↵ers from the classical results. Lastly, we investigate the statistical mechanics of the Sadowsky ribbon in a sim- ilar spirit to the famous Kratky-Porod model. We present a detail theoretical and numerical calculations of the Sadowsky ribbon under the e↵ect of external force and torsion. This model may be able to explain new and novel biopolymers ranging from actin, microtubules to rod-like viruses that lies outside the scope of WLC model. This concludes the thesis. Contents TitlePage.................................... i Abstract..................................... iii TableofContents................................ v ListofFigures.................................. viii List of Tables . xi Citations to Previously Published Work . xii Acknowledgments................................ xiii Dedication . xvi 1 Introduction, Motivation, and Outline 1 1.1 Outline of Thesis . 4 1.2 Role of defects in the crumpling of crystalline shell . 5 1.3 Spores and buckled elastica . 7 1.4 Chapter 4: Statistical Mechanics of Twisted Ribbons . 8 2 Buckling morphologies of crystalline shells with frozen defects 11 2.1 Introduction . 11 2.2 Elastic Theory . 18 2.3 Topological defects on crystalline shells . 21 2.3.1 Set of minimal topological defects . 21 2.3.2 Spheres with one type of disclination . 23 2.3.3 Spheres with more than one type of disclinations . 26 2.3.4 Sphere with positive and negative disclinations . 28 2.4 Elastic energies of thin crystalline shells . 28 2.5 Phase space . 31 2.6 Classification of shell type . 37 2.7 ShapeAnalysis .............................. 39 2.7.1 Group Theory . 39 2.7.2 Comparison with crumpled spheres . 41 2.7.3 “Shape” Spectroscopy . 43 2.8 Hysteresis ................................. 48 v Contents vi 2.8.1 icosahedral-sphere . 50 2.8.2 dodecahedral-sphere . 53 2.8.3 cubical-sphere and octahedral-sphere . 56 2.9 Landau Theory . 61 2.9.1 Order Parameter . 61 2.9.2 Simplest Landau model . 62 2.9.3 Coupling between Order Parameters . 65 2.10 Conclusion . 70 3 Physical basis of the bacterial spore coat architecture and its relation to the elastic 73 3.1 Introduction . 73 3.2 NumericalSimulations .......................... 78 3.3 Discussions . 82 3.4 Mathematical Model . 86 3.5 BifurcationAnalysis ........................... 90 3.5.1 Linearized Problem . 91 3.5.2 Stability of Fundamental Solution . 93 3.5.3 Phase Diagram . 95 3.6 Recognition problem . 98 3.6.1 Steady-State bifurcation . 98 3.6.2 Hopfbifurcation ......................... 104 3.7 conclusion . 105 4 Statistical Mechanics of Twisted Ribbons 109 4.1 Introduction . 109 4.2 Physical Model . 111 4.3 Ribbon Theory . 114 4.4 NumericalMethods............................ 118 4.5 Theoretical Methods . 122 4.6 ResultsandDiscussions . 127 4.7 Conclusion . 132 A Continuum-Elastic Theory 134 A.1 F¨oppl-vonK´arm´an-Donnellequations . 135 A.2 LinearBucklingAnalysis. 140 B Surface Evolver 144 B.1 Elastic energy . 144 B.2 Triangulation . 145 B.3 Energy of thin Shell . 146 Contents vii C Mathematical description of Platonic solids 148 C.1 Onaka’s formulation . 148 D Information on Bacterial Spores 151 D.1 Surface topography and height measurements of bacterial spores . 151 D.2 Identification of coat proteins not required for significant coat sti↵ness 151 E Mathematical background on Bifurcation 154 E.1 Implicit Function Theorem . 154 E.2 FredholmAlternative. 155 E.3 Liapunov-Schmidt reduction method . 156 FReviewofDNAmodels 161 F.1 RandomWalkModel........................... 161 F.2 FreelyJointChain ............................ 162 F.3 Worm-LikeChainModel . 163 G Sadowsky Functional 170 H Twist and Writhe 173 H.1 Twist.................................... 174 H.2 NumericalimplementationofWrithe . 175 Bibliography 178 List of Figures 1.1 Crumpling occurs on a wide range of length scale . 3 2.1 Di↵erence between frozen topography and frozen defects . 16 2.2 Sequential deformation of a thin Icosahedron type shell and Dodeca- hedrontypeshell ............................. 24 2.3 Deformation of an Antiprism-4 type shell, a Pyramid-5 type shell, a Dipyramid-3 type shell and a Prism-7 type shell . 27 2.4 Plot of Energy vs Volume as the sphere shrinks for di↵erent values of h/R .................................... 29 2.5 Comparison of the extend of buckling for shell with same symmetry group G but di↵erent CK coordinates . 31 2.6 The morphological phase diagram of sphere with icosahedral defect (M1)anddodecahedraldefect(M2)asafunctionoftheaspectratio h/R and lattice ratio R/a ........................ 33 2.7 The morphological phase diagram of sphere with octahedral defect (M3)andcubicaldefect(M4)asafunctionoftheaspectratioh/R and lattice ratio R/a ........................... 34 2.8 The morphological phase diagram of sphere with tetrahedral defect (M5)asafunctionoftheaspectratioh/R and lattice ratio R/a.. 35 2.9 Phase diagram as a function of h/R and R/a .............. 36 2.10 Comparing shells formed from an octahedron (M3)andacube(M4). 38 2.11 Crumpled shells with volume shrinkage 20% . 41 2.12 P-V response of a crushed icosahedral-sphere and the plot of the nor- malized Qˆ` and Wˆ ` of the icosahedral-sphere during the Loading- Unloadingcycle..............................| | 52 2.13 P-V response of a crushed dodecahedral-sphere and the plot of the normalized Qˆ` and Wˆ ` of the dodecahedral-sphere during the Loading- Unloading(LU)cycle...........................| | 54 viii List of Figures ix 2.14 P-V response of a crushed cubical-sphere and the plot of the normalized Qˆ` and Wˆ ` of the cubical-sphere during the Loading-Unloading (LU) cycle....................................| | 58 2.15 P-V response of a crushed octahedral-sphere and the plot of the nor- malized Qˆ` and Wˆ ` of the octahedral-sphere during the Loading- Unloading(LU)cycle...........................| | 59 2.16 The coupling term is plotted for the various points along the loading- unloading cycle of the icosahedral-sphere. 69 3.1 Morphology of the B. subtilis spore . 75 3.2 Ruck formation in B. subtilis ...................... 77 3.3 Finite range hookean restoring force between spore coat and core sub- strate.................................... 80 3.4 SchematicoftheSporemodel . 81 3.5 Model of formation of folds in the spore coat and their response to spore swelling . 84 3.6 Deformation of a beam cross-section . 87 3.7 Stability diagram of the elastic beam . 94 3.8 Phase diagram showing when det = 0 or trace = 0 . 96 3.9 Complete phase diagram of the applied load versus slenderness for µ =0.1and` =1............................. 97 3.10 Term in the normal form, guuu± as a function of slenderness λ ..... 101 3.11 Bifurcation diagrams associated with the buckling of the compressible beam on elastic spring support as a function of the applied force P .103 3.12 Plot of first Lyapunov coefficient as a function of slenderness . 106 4.1 Schematic of imaginary loop added to the DNA chain to form a closed loop . 117 4.2 Schematicofthediscretechain . 118 4.3 Simulated extension of a torque-free ribbon at di↵erent normalized temperatures . 127 4.4 Force versus extension curves of Sadowsky ribbon . 128 4.5 Force-Extension curve for a Sadowsky ribbon at fixed temperature un- der di↵erent applied torque . 130 4.6 Relative extension of ribbon versus excess linking number at di↵erent fixed applied force and normalized temperature . 131 B.1 Triangulation of a surface . 145 B.2 Plot of Energy vs R/a for the platonic solids for h/R = 0.15 . 147 C.1 The Platonic solids plotted using Onaka’s formulas .
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