DOCSLIB.ORG

PDF (Alberto Leonardi's Phd Thesis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
PDF (Alberto Leonardi's Phd Thesis XXIV cycle Doctoral School in Materials Science and Engineering Molecular Dynamics and X-ray Powder Diffraction Simulations Investiigatiion of nano-pollycrystalllliine miicrostructure at the atomiic scalle couplliing llocall structure confiiguratiions and X-ray Powder Diiffractiion techniiques Alberto Leonardi November 2012 MOLECULAR DYNAMICS AND X-RAY POWDER DIFFRACTION SIMULATIONS INVESTIGATION OF NANO-POLYCRYSTALLINE MICROSTRUCTURE AT THE ATOMIC SCALE COUPLING LOCAL STRUCTURE CONFIGURATIONS AND X-RAY POWDER DIFFRACTION TECHNIQUES Alberto Leonardi E-mail: [email protected] Approved by: Ph.D. Commission: Prof. Paolo Scardi, Advisor Prof. Alberto Quaranta Department of Materials Engineering Department of Materials Engineering and Industrial Technologies and Industrial Technologies University of Trento, Italy. University of Trento, Italy. Prof. Matteo Leoni Prof. Rozalya Barabash Department of Materials Engineering Materials Science & Technology Div. and Industrial Technologies Oak Ridge National Laboratory, University of Trento, Italy. Tenneesee (US). Prof. Marco Milanesio Department of Science and advanced technologies University of Piemonte Orientale, Italy. University of Trento, faculty of engineering Department of Materials Engineering and Industrial Technologies November 2012 University of Trento - Department of Materials Engineering and Industrial Technologies Doctoral Thesis Alberto Leonardi - 2012 Published in Trento (Italy) – by University of Trento ISBN: 978-88-8443-455-5 "The essence of science lies not in discovering facts, but in discovering new ways of thinking about them. “ W. L. Bragg Abstract Atomistic simulations based on Molecular Dynamics (MD) were used to model the lattice distortions in metallic nano-polycrystalline microstructures, with the purpose of supporting the analysis of the X-ray powder diffraction patterns with a better, atomic level understanding of the studied system. Complex microstructures were generated with a new modified Voronoi tessellation method which provides a direct relation between generation parameters and statistical properties of the resulting model. MD was used to equilibrate the system: the corresponding strain field was described both in the core and in surface regions of the different crystalline domains. New methods were developed to calculate the strain tensor at the atomic scale. Line Profile Analysis (LPA) was employed to retrieve the microstructure information (size and strain effects) from the powder diffraction patterns: a general algorithm with an atomic level resolution was developed to consider the size effects of crystalline domains of any arbitrary shape. The study provided a new point of view on the role of the grain boundary regions in nano-polycrystalline aggregates, exploring the interference effects between different domains and between grain boundary and crystalline regions. Usual concepts of solid mechanics were brought in the atomistic models to describe the strain effects on the powder diffraction pattern. To this purpose the new concept of Directional - Pair Distribution Function (D-PDF) was developed. D-PDFs calculated from equilibrated atomistic simulations provide a representation of the strain field which is directly comparable with the results of traditional LPA (e.g. Williamson-Hall plot and Warren-Averbach method). The D-PDF opens a new chapter in powder diffraction as new insights and a more sound interpretation of the results are made possible with this new approach to diffraction LPA. Table of contents Chapter I Introduction .............................................................................................. 13 Chapter II Modelling of Material Microstructures ................................................. 17 2.1. Abstract ....................................................................................... 17 2.2. Introduction ................................................................................. 18 2.3. Methods ....................................................................................... 20 2.3.1. Traditional stochastic tessellation methods .......................... 20 2.3.2. Modified Voronoi Tessellation (MVT) .................................... 22 2.3.2.1. Relationships between traditional tessellations and the MVT .......................................................................................... 25 2.3.3. Constrained Modified Voronoi Tessellation (CMVT) ............ 27 2.4. Results and discussion ............................................................. 28 2.4.1. Atomic density and voids in MVT-derived microstructures .. 28 2.4.2. Statistical properties of the MVT ........................................... 32 2.4.3. Relationship between input parameters and resulting microstructure ........................................................................ 36 2.4.4. Reliability of MVT statistics by the evolutionary CMVT ........ 39 2.4.5. Multiple target properties optimization with CMVT ............... 41 2.4.6. MVT computing performance ................................................ 44 2.5. Conclusion .................................................................................. 46 Chapter III Analysis of Atomistic Simulation Data ................................................ 47 3.1. Abstract ....................................................................................... 47 3.2. Introduction ................................................................................. 48 3.3. Methods ....................................................................................... 49 3.3.1. Molecular Dynamics simulation ............................................. 49 3.3.2. Global mean square displacement (global MSD) ................. 51 9 3.3.3. Local coordination and surface shape .................................. 52 3.3.4. Strain at the atomic level ....................................................... 54 3.3.4.1. The Voronoi Cell Deformation method (VCD) ................ 55 3.3.4.2. The evolutional Voronoi Cell Deformation method (eVCD) .......................................................................................... 56 3.3.5. Isotropic and Anisotropic strains ........................................... 57 3.3.6. Potential energy and Stress at the atomic level ................... 59 3.4. Results and discussion ............................................................. 60 3.4.1. Strain at the atomic level in a nano-polycrystalline microstructure from MD .................................................................................. 60 3.4.2. Stress – Strain relation in polycrystalline microstructure ..... 68 3.4.3. Preliminary X-ray Diffraction Line Profile analysis ............... 70 3.5. Conclusion .................................................................................. 73 3.6. Appendix III.A: Deformation of convex polyhedron from volume properties ...................................................................................................... 74 3.7. Appendix III.B: Deformation of convex polyhedron from mass properties ...................................................................................................... 76 Chapter IV Interference Effects in Nano-crystalline Systems .............................. 77 4.1. Abstract ....................................................................................... 77 4.2. Introduction ................................................................................. 78 4.3. Generation of the nano-polycrystalline model ...................... 78 4.4. Results and Discussion ............................................................ 79 4.5. Conclusions ................................................................................ 86 Chapter V Common Volume Function & Diffraction Line Profiles ..................... 87 5.1. Abstract ....................................................................................... 87 5.2. Introduction ................................................................................. 88 5.3. Common Volume Function of polyhedral crystallites .......... 90 10 5.3.1. Convex polyhedra .................................................................. 90 5.3.2. Non-convex polyhedra ........................................................... 93 5.4. Examples of application ............................................................ 93 5.4.1. Convex shapes ...................................................................... 94 5.4.1.1. Truncated and bitruncated cube ..................................... 94 5.4.1.2. Irregular domain shapes: 3D Voronoi cell ...................... 95 5.4.1.3. Polycrystalline microstructure: 3D Poisson-Voronoi microstructure .................................................................. 97 5.4.2. Non-convex shapes ............................................................... 98 5.4.2.1. Planar tripods and tetrapods ........................................... 98 5.4.2.2. Hollow cubes ................................................................... 100 5.4.3. Non-polyhedral shapes .......................................................... 102 5.5. Conclusions ................................................................................ 104 5.6. Appendix V.A: Common Volume Function in the frame of set theory ......... 105 Chapter VI Directional - Pair Distribution Function ............................................... 107 6.1. Abstract ......................................................................................
Load more

Recommended publications
  • Archimedean Solids
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2008 Archimedean Solids Anna Anderson University of Nebraska-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Anderson, Anna, "Archimedean Solids" (2008). MAT Exam Expository Papers. 4. https://digitalcommons.unl.edu/mathmidexppap/4 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Archimedean Solids Anna Anderson In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. Jim Lewis, Advisor July 2008 2 Archimedean Solids A polygon is a simple, closed, planar figure with sides formed by joining line segments, where each line segment intersects exactly two others. If all of the sides have the same length and all of the angles are congruent, the polygon is called regular. The sum of the angles of a regular polygon with n sides, where n is 3 or more, is 180° x (n – 2) degrees. If a regular polygon were connected with other regular polygons in three dimensional space, a polyhedron could be created. In geometry, a polyhedron is a three- dimensional solid which consists of a collection of polygons joined at their edges. The word polyhedron is derived from the Greek word poly (many) and the Indo-European term hedron (seat).
    [Show full text]
  • A Fun Approach to Teaching Geometry and Inspiring Creativity
    Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture A Fun Approach to Teaching Geometry and Inspiring Creativity Ioana Browne, Michael Browne, Mircea Draghicescu*, Cristina Draghicescu, Carmen Ionescu ITSPHUN LLC 2023 NW Lovejoy St. Portland, OR 97209, USA [email protected] Abstract ITSPHUNTM geometric shapes can be easily combined to create an infinite number of structures both decorative and functional. We introduce this system of shapes, and discuss its use for teaching mathematics. At the workshop, we will provide a large number of pieces of various shapes, demonstrate their use, and give all participants the opportunity to build their own creations at the intersection of art and mathematics. Introduction Connecting pieces of paper by sliding them together along slits or cuts is a well-known method of building 3D objects ([2], [3], [4]). Based on this idea we developed a system of shapes1 that can be used to build arbitrarily complex structures. Among the geometric objects that can be built are Platonic, Archimedean, and Johnson solids, prisms and antiprisms, many nonconvex polyhedra, several space-filling tessellations, etc. Guided by an user’s imagination and creativity, simple objects can then be connected to form more complex constructs. ITSPHUN pieces made of various materials2 can be used for creative play, for making functional and decorative objects such as jewelry, lamps, furniture, and, in classroom, for teaching mathematical concepts.3 1Patent pending. 2We used many kinds of paper, cardboard, foam, several kinds of plastic, wood and plywood, wood veneer, fabric, and felt. 3In addition to models of geometric objects, ITSPHUN shapes have been used to make birds, animals, flowers, trees, houses, hats, glasses, and much more - see a few examples in the photo gallery at www.itsphun.com.
    [Show full text]
  • Uniform Panoploid Tetracombs
    Uniform Panoploid Tetracombs George Olshevsky TETRACOMB is a four-dimensional tessellation. In any tessellation, the honeycells, which are the n-dimensional polytopes that tessellate the space, Amust by definition adjoin precisely along their facets, that is, their ( n!1)- dimensional elements, so that each facet belongs to exactly two honeycells. In the case of tetracombs, the honeycells are four-dimensional polytopes, or polychora, and their facets are polyhedra. For a tessellation to be uniform, the honeycells must all be uniform polytopes, and the vertices must be transitive on the symmetry group of the tessellation. Loosely speaking, therefore, the vertices must be “surrounded all alike” by the honeycells that meet there. If a tessellation is such that every point of its space not on a boundary between honeycells lies in the interior of exactly one honeycell, then it is panoploid. If one or more points of the space not on a boundary between honeycells lie inside more than one honeycell, the tessellation is polyploid. Tessellations may also be constructed that have “holes,” that is, regions that lie inside none of the honeycells; such tessellations are called holeycombs. It is possible for a polyploid tessellation to also be a holeycomb, but not for a panoploid tessellation, which must fill the entire space exactly once. Polyploid tessellations are also called starcombs or star-tessellations. Holeycombs usually arise when (n!1)-dimensional tessellations are themselves permitted to be honeycells; these take up the otherwise free facets that bound the “holes,” so that all the facets continue to belong to two honeycells. In this essay, as per its title, we are concerned with just the uniform panoploid tetracombs.
    [Show full text]
  • 15 BASIC PROPERTIES of CONVEX POLYTOPES Martin Henk, J¨Urgenrichter-Gebert, and G¨Unterm
    15 BASIC PROPERTIES OF CONVEX POLYTOPES Martin Henk, J¨urgenRichter-Gebert, and G¨unterM. Ziegler INTRODUCTION Convex polytopes are fundamental geometric objects that have been investigated since antiquity. The beauty of their theory is nowadays complemented by their im- portance for many other mathematical subjects, ranging from integration theory, algebraic topology, and algebraic geometry to linear and combinatorial optimiza- tion. In this chapter we try to give a short introduction, provide a sketch of \what polytopes look like" and \how they behave," with many explicit examples, and briefly state some main results (where further details are given in subsequent chap- ters of this Handbook). We concentrate on two main topics: • Combinatorial properties: faces (vertices, edges, . , facets) of polytopes and their relations, with special treatments of the classes of low-dimensional poly- topes and of polytopes \with few vertices;" • Geometric properties: volume and surface area, mixed volumes, and quer- massintegrals, including explicit formulas for the cases of the regular simplices, cubes, and cross-polytopes. We refer to Gr¨unbaum [Gr¨u67]for a comprehensive view of polytope theory, and to Ziegler [Zie95] respectively to Gruber [Gru07] and Schneider [Sch14] for detailed treatments of the combinatorial and of the convex geometric aspects of polytope theory. 15.1 COMBINATORIAL STRUCTURE GLOSSARY d V-polytope: The convex hull of a finite set X = fx1; : : : ; xng of points in R , n n X i X P = conv(X) := λix λ1; : : : ; λn ≥ 0; λi = 1 : i=1 i=1 H-polytope: The solution set of a finite system of linear inequalities, d T P = P (A; b) := x 2 R j ai x ≤ bi for 1 ≤ i ≤ m ; with the extra condition that the set of solutions is bounded, that is, such that m×d there is a constant N such that jjxjj ≤ N holds for all x 2 P .
    [Show full text]
  • Snubs, Alternated Facetings, & Stott-Coxeter-Dynkin Diagrams
    Symmetry: Culture and Science Vol. 21, No.4, 329-344, 2010 SNUBS, ALTERNATED FACETINGS, & STOTT-COXETER-DYNKIN DIAGRAMS Dr. Richard Klitzing Non-professional mathematician, (b. Laupheim, BW., Germany, 1966). Address: Kantstrasse 23, 89522 Heidenheim, Germany. E-mail: [email protected] . Fields of interest: Polytopes, geometry, mathematical quasi-crystallography. Publications: (2002) Axial Symmetrical Edge-Facetings of Uniform Polyhedra. In: Symmetry: Cult. & Sci., vol. 13, nos. 3- 4, pp. 241-258. (2001) Convex Segmentochora. In: Symmetry: Cult. & Sci., vol. 11, nos. 1-4, pp. 139-181. (1996) Reskalierungssymmetrien quasiperiodischer Strukturen. [Symmetries of Rescaling for Quasiperiodic Structures, Ph.D. Dissertation, in German], Eberhard-Karls-Universität Tübingen, or Hamburg, Verlag Dr. Kovač, ISBN 978-3-86064-428- 7. – (Former ones: mentioned therein.) Abstract: The snub cube and the snub dodecahedron are well-known polyhedra since the days of Kepler. Since then, the term “snub” was applied to further cases, both in 3D and beyond, yielding an exceptional species of polytopes: those do not bow to Wythoff’s kaleidoscopical construction like most other Archimedean polytopes, some appear in enantiomorphic pairs with no own mirror symmetry, etc. However, they remained stepchildren, since they permit no real one-step access. Actually, snub polytopes are meant to be derived secondarily from Wythoffians, either from omnitruncated or from truncated polytopes. – This secondary process herein is analysed carefully and extended widely: nothing bars a progress from vertex alternation to an alternation of an arbitrary class of sub-dimensional elements, for instance edges, faces, etc. of any type. Furthermore, this extension can be coded in Stott-Coxeter-Dynkin diagrams as well.
    [Show full text]
  • Convex Polytopes and Tilings with Few Flag Orbits
    Convex Polytopes and Tilings with Few Flag Orbits by Nicholas Matteo B.A. in Mathematics, Miami University M.A. in Mathematics, Miami University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Abstract of Dissertation The amount of symmetry possessed by a convex polytope, or a tiling by convex polytopes, is reflected by the number of orbits of its flags under the action of the Euclidean isometries preserving the polytope. The convex polytopes with only one flag orbit have been classified since the work of Schläfli in the 19th century. In this dissertation, convex polytopes with up to three flag orbits are classified. Two-orbit convex polytopes exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, the rhombic dodecahedron, and the rhombic triacontahedron. Two-orbit face-to-face tilings by convex polytopes exist on E1, E2, and E3; the only ones which are also combinatorially two-orbit are the trihexagonal plane tiling, the rhombille plane tiling, the tetrahedral-octahedral honeycomb, and the rhombic dodecahedral honeycomb. Moreover, any combinatorially two-orbit convex polytope or tiling is isomorphic to one on the above list. Three-orbit convex polytopes exist in two through eight dimensions. There are infinitely many in three dimensions, including prisms over regular polygons, truncated Platonic solids, and their dual bipyramids and Kleetopes. There are infinitely many in four dimensions, comprising the rectified regular 4-polytopes, the p; p-duoprisms, the bitruncated 4-simplex, the bitruncated 24-cell, and their duals.
    [Show full text]
  • Separators of Simple Polytopes
    Fachbereich Mathematik und Informatik der Freien Universit¨atBerlin Separators of simple polytopes A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Natural Sciences to the Department of Department of Mathematics and Computer Science by Lauri Loiskekoski Berlin 2017 Supervisor: Prof. Dr. G¨unter M. Ziegler Second Examiner: Prof. Dr. Volker Kaibel Date of defense: 21.12.2017 Acknowledgements First and foremost I would like to thank my supervisor G¨unter M. Ziegler for introducing me to this fascinating topic and always having time to discuss. You always knew the right references and had good ideas what to look at. I would like to thank Hao Chen, Arnau Padrol and Raman Sanyal for help- ing me get started on research and and answering my questions on separators and polytopes. I would like to thank my friends and colleague at the villa, in particular Francesco Grande, Katy Beeler, Albert Haase, Tobias Friedl, Philip Brinkmann, Nevena Pali´c,Jean-Philippe Labb´e,Giulia Codenotti, Jorge Alberto Olarte, Hannah Sch¨aferSj¨oberg and Anna Maria Hartkopf. Thanks for navigating the bureaucracy go to Elke Pose. Thanks to Johanna Steinmeyer for the TikZ pictures. I wish to thank the Berlin Mathematical School for funding my research and European Research Council and Osaka University for funding my travels. Finally, I would like to thank my friends in Finland, especially the people of Rissittely for answering all the unconventional questions I have had about academia and life in general. iii iv Contents Acknowledgements . iii 1 Introduction 1 2 Graphs and Polytopes 3 2.1 Polytopes .
    [Show full text]
  • Numerically Stable Form Factor of Any Polygon and Polyhedron
    research papers Numerically stable form factor of any polygon and polyhedron ISSN 1600-5767 Joachim Wuttke* Forschungszentrum Ju¨lich GmbH, Ju¨lich Centre for Neutron Science (JCNS) at Heinz Maier-Leibnitz Zentrum (MLZ), Lichtenbergstrasse 1, 85748 Garching, Germany. *Correspondence e-mail: [email protected] Received 15 January 2021 Accepted 11 February 2021 Coordinate-free expressions for the form factors of arbitrary polygons and polyhedra are derived using the divergence theorem and Stokes’s theorem. Apparent singularities, all removable, are discussed in detail. Cancellation near Edited by D. I. Svergun, European Molecular the singularities causes a loss of precision that can be avoided by using series Biology Laboratory, Hamburg, Germany expansions. An important application domain is small-angle scattering by nanocrystals. Keywords: form factors; polyhedra; Fourier shape transform. 1. Introduction 1.1. Overview The term ‘form factor’ has different meanings in science and in engineering. Here, we are concerned with the form factor of a geometric figure as defined in the physical sciences, namely the Fourier transform of the figure’s indicator function, also called the ‘shape transform’ of the figure. This form factor has important applications in the emission, detection and scattering of radiation. Two-dimensional shape transforms are used in the theory of reflector antennas (Lee & Mittra, 1983). The three-dimensional form factors of the sphere and the cylinder go back to Lord Rayleigh (1881). Shapes of three-dimensional nanoparticles are investigated by neutron and X-ray small-angle scattering (Hammouda, 2010). Particles grown on a substrate (Henry, 2005) develop many different shapes, especially polyhedral ones, as observed by grazing-incidence neutron and X-ray small-angle scattering (GISAS, GISANS, GISAXS) (Renaud et al., 2009).
    [Show full text]
  • Optimization Methods in Discrete Geometry
    Optimization Methods in Discrete Geometry Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) am Fachbereich Mathematik und Informatik der Freien Universität Berlin von Moritz Firsching Berlin 2015 Betreuer: Prof. Günter M. Ziegler, PhD Zweitgutachter: Prof. Dr. Dr.Jürgen Richter-Gebert Tag der Disputation: 28. Januar 2016 וראיתי תשוקתך לחכמות הלמודיות עצומה והנחתיך להתלמד בהם לדעתי מה אחריתך. 4 (דלאלה¨ אלחאירין) Umseitiges Zitat findet sich in den ersten Seiten des Führer der Unschlüssigen oder kurz RaMBaM רבי משה בן מיימון von Moses Maimonides (auch Rabbi Moshe ben Maimon des ,מורה נבוכים ,aus dem Jahre 1200. Wir zitieren aus der hebräischen Übersetzung (רמב״ם judäo-arabischen Originals von Samuel ben Jehuda ibn Tibbon aus dem Jahr 1204. Hier einige spätere Übersetzungen des Zitats: Tunc autem vidi vehementiam desiderii tui ad scientias disciplinales: et idcirco permisi ut exerceres anima tuam in illis secundum quod percepi de intellectu tuo perfecto. —Agostino Giustiniani, Rabbi Mossei Aegyptii Dux seu Director dubitantium aut perplexorum, 1520 Und bemerkte ich auch, daß Dein Eifer für das mathematische Studium etwas zu weit ging, so ließ ich Dich dennoch fortfahren, weil ich wohl wußte, nach welchem Ziele Du strebtest. — Raphael I. Fürstenthal, Doctor Perplexorum von Rabbi Moses Maimonides, 1839 et, voyant que tu avais un grand amour pour les mathématiques, je te laissais libre de t’y exercer, sachant quel devait être ton avenir. —Salomon Munk, Moise ben Maimoun, Dalalat al hairin, Les guide des égarés, 1856 Observing your great fondness for mathematics, I let you study them more deeply, for I felt sure of your ultimate success.
    [Show full text]
  • Polygloss V0.05
    PolyGloss PolyGloss v0.05 This is my multidimensional glossary. This is a cut on the terms as viewed by someone who can visualise four and higher dimensions, and finds the commonly used terms to be more of a hinderance. Normal Entry A normal entry appears in this style. The header appears in normal bold font. Comment A comment appears with the header in italics, such as this entry. Comments are not suggested for use based on this glossary, but are more in line to allow the reader to find the preferred entry. Decimal (dec) Numbers written in groups of three, or continuous, and using a full stop, are decimal numbers. Each column is 10 times the next. If there is room for doubt, the number is prefixed with "dec", eg dec 288 = twe 248. There are no special symbols for the teenties (V0) or elefties (E0), as these carry. twe V0 = dec 100, twe E0 = dec 110. The letter E is used for an exponent, but of 10, not 100, so 1E5 is 100 000, not 10 000 000 000, or 235. ten 10 10 ten million 10 000000 594 5340 hundred 100 V0 100 million 100 000000 57V4 5340 thousand 1000 840 1 milliard 1000 000000 4 9884 5340 10 thous 10000 8340 100 thous 100000 6 E340 1 million 1000000 69 5340 Twelfty (twe) Numbers written in groups of two or four, and using a colon (:) for a radix are numbers in base 120. Pairs of digits make up a single place, the first is 10 times the second: so in 140: = 1 40, the number is 1*120+40.
    [Show full text]
  • Double-Link Expandohedra: a Mechanical Model for Expansion of a Virus
    Double-link expandohedra: a mechanical model for expansion of a virus By F. Kovacs´ 1, T. Tarnai2, S.D. Guest3 and P.W. Fowler4 1Research Group for Computational Structural Mechanics, Hungarian Academy of Sciences, Budapest, M˝uegyetem rkp. 3., H-1521 Hungary 2Department of Structural Mechanics, Budapest University of Technology and Economics, Budapest, M˝uegyetem rkp. 3., H-1521 Hungary 3Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK 4Department of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, UK Double-link expandohedra are introduced: each is constructed from a parent poly- hedron by replacing all faces with rigid plates, adjacent plates being connected by a pair of spherically jointed bars. Powerful symmetry techniques are developed for mobility analysis of general double-link expandohedra, and when combined with numerical calculation and physical model building, demonstrate the existence of generic finite breathing expansion motions in many cases. For icosahedrally sym- metric trivalent parents with pentagonal and hexagonal faces only (fullerene poly- hedra), the derived expandohedra provide a mechanical model for the experimen- tally observed swelling of viruses such as cowpea chlorotic mottle virus (CCMV). A fully symmetric swelling motion (a finite mechanism) is found for systems based on icosahedral fullerene polyhedra with odd triangulation number, T ≤ 31, and is conjectured to exist for all odd triangulation numbers. Keywords: Symmetry; Polyhedra; Mechanism; Virus Structures 1. Introduction Many viruses consist of an outer protein coat (the virion) containing a DNA or RNA ‘payload’, where the virion undergoes reversible structural changes that allow switchable access to the interior by the opening of interstices through expansion.
    [Show full text]
  • Arxiv:2103.12769V1 [Math.MG] 23 Mar 2021
    MONO-MONOSTATIC POLYHEDRA WITH UNIFORM POINT MASSES HAVE AT LEAST 8 VERTICES SANDOR´ BOZOKI,´ GABOR´ DOMOKOS, FLORI´ AN´ KOVACS´ AND KRISZTINA REGOS} Abstract. The monostatic property of convex polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in the focus of research ever since Conway and Guy [3] published the proof of the existence of the first such object, followed by the constructions of Bezdek [2] and Reshetov [16]. These examples establish F ≤ 14;V ≤ 18 as the respective upper bounds for the minimal number of faces and vertices for a homogeneous mono-stable polyhedron. By proving that no mono-stable homogeneous tetra- hedron existed, Conway and Guy [3] established for the same problem the lower bounds for the number of faces and vertices as F; V ≥ 5 and the same lower bounds were also established [9] for the mono-unstable case. It is also clear that the F; V ≥ 5 bounds also apply for convex, homogeneous point sets with unit masses at each point (also called polyhedral 0-skeletons) and they are also valid for mono-monostatic polyhedra with exactly on stable and one unstable equilibrium point (both homogeneous and 0-skeletons). Here we present an algorithm by which we improve the lower bound to V ≥ 8 vertices (implying f ≥ 6 faces) on mono-unstable and mono-monostable 0-skeletons. Our algorithm appears to be less well suited to compute the lower bounds for mono-stability. We point out these difficulties in connection with the work of Dawson and Finbow [4, 5, 6] who explored the monostatic property of simplices in higher dimensions.
    [Show full text]