DOCSLIB.ORG

Density-Functional Theory with Screened Van Der Waals Interactions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Density-Functional Theory with Screened Van Der Waals Interactions Adsorption of Atoms and Molecules on Surfaces: Density-Functional Theory with Screened van der Waals Interactions VICTOR GONZALO RUIZ LÓPEZ BERLIN 2016 • Adsorption of Atoms and Molecules on Surfaces: Density-Functional Theory with Screened van der Waals Interactions vorgelegt von M.Sc. Victor Gonzalo Ruiz López geb. in Mexiko Stadt der Fakultät II Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr.rer.nat genehmigte Dissertation Promotionsausschuss: Vorsitzender: Prof. Dr. Mario Dähne 1. Gutachter: Prof. Dr. Eckehard Schöll 2. Gutachter: Prof. Dr. Alexandre Tkatchenko Tag der wissenschaftlichen Aussprache: 18. Mai 2016 Berlin – 2016 This thesis is licensed under a Creative Commons Attribution-ShareAlike 4.0 Interna- tional License (Free Culture License): https://creativecommons.org/licenses/by-sa/4.0/. Abstract Understanding the properties of novel hybrid interfaces has implications in both fun- damental science and technology. The interfaces of hybrid inorganic/organic systems (HIOS), for instance, may lead to the emergence of collective effects that the isolated components forming the interface do not exhibit. The electronic properties and the function of these interfaces are intimately linked to their interface geometry. The in- terface structure and its properties are a result of the interplay of electron transfer pro- cesses, (covalent) hybridization of wave functions, van der Waals (vdW) interactions, and Pauli repulsion. In particular, vdW interactions are fundamental in determining the structure of the interface and the stability of HIOS. Thus, controlling the functionalities of HIOS involves as a first step the prediction and understanding of the structural fea- tures of the interface. Since the role of vdW interactions is crucial in the determination of the structural features and stability, their accurate prediction becomes distinctively relevant in this context. From an atomistic perspective within the computational simu- lation of materials, it follows that modeling the adsorption of atoms and molecules on surfaces requires efficient electronic-structure methods that are able to capture both co- valent and non-covalent interactions in a reliable manner. It is in this context that we propose a method within density-functional approxi- mations (DFA) which includes screened vdW interactions (the so-called DFA+vdWsurf method) to model the adsorption of atoms and molecules on surfaces. Specifically, we combine dispersion-corrected density-functional approximations (the Tkatchenko- Scheffler DFA+vdW method) with the Lifshitz-Zaremba-Kohn theory in order to include the Coulomb screening within the substrate surface in the determination of the vdW C6 coefficients and vdW radii. Our method includes both image-plane and interface po- larization effects via the inclusion of semi-local hybridization due to the dependence of ab the C6 interatomic parameters on the electron density. We show that the inclusion of the non-local many-body response of the substrate electrons is essential to predict the structure and stability of atoms and molecules on surfaces in a reliable manner, taking as examples the adsorption of a Xe monolayer and an aromatic molecule which includes oxygen in its structure. In particular, we show that the DFA+vdWsurf method yields geometries in remark- able agreement (within approximately 0.1 Å) with normal incidence x-ray standing wave measurements for the adsorption of 3,4,9,10-perylene-tetracarboxylic dianhydride (C24H8O6, PTCDA) on Ag(111), Au(111), and Cu(111). Additional examples include the i ii adsorption of Xe on transition-metal surfaces and a comparative study of the interfaces formed by a PTCDA monolayer on the Ag(111), Ag(100), and Ag(110) surfaces, show- ing that the method also achieves surface-termination sensitivity. These results demon- strate that the DFA+vdWsurf method can deal in a reliable manner with a wide range of interactions at HIOS including chemical interactions, electrostatic interactions, Pauli re- pulsion, and vdW interactions; therefore establishing it as a reasonable option for the ac- curate treatment of adsorption problems due to its efficiency and affordability in terms of computational time. Nevertheless, cooperative electronic effects between atoms in larger molecules lead to non-additive molecular polarizabilities, effect which is absent in the DFA+vdWsurf method. I conclude by commenting this and further remaining chal- lenges left in order to achieve both quantitative accuracy and predictive power in the simulation of the structure and stability of complex interfaces. Zusammenfassung Das Verständnis der Eigenschaften von neuartigen Hybrid-Grenzflächen ist sowohl für die Grundlagenforschung als auch für technologische Anwendungen wichtig. Die Grenzflächen von hybriden anorganischen/organischen Systemen (HIOS) führen zum Beispiel zur Entstehung von kollektiven Effekten, welche die isolierten Bausteine nicht besitzen. Die elektronischen Eigenschaften und die Funktionen dieser Grenzflächen hängen stark von ihrer Geometrie ab. Die Grenzflächenstruktur und ihre Eigenschaften resultieren aus dem Wechselspiel zwischen Elektronentransferprozessen, (kovalenter) Hybridisierung von Wellenfunktionen, van-der-Waals-Wechselwirkungen (vdW) und der Pauli-Abstoßung. Die vdW-Wechselwirkungen sind entscheidend für die Struktur und die Stabilität der HIOS. Um schlussendlich die Funktionalitäten der HIOS kon- trollieren zu können, ist als erster Schritt die Vorhersage und das Verständnis von strukturellen Merkmalen der Grenzfläche notwendig. Aufgrund der bedeutenden Rolle der vdW-Wechselwirkungen in diesem Kontext ist es besonders wichtig diese Wechsel- wirkungen präzise berechnen zu können. Von einer atomistischen Perspektive benötigt die Computersimulation der Adsorption von Atomen und Molekülen auf Oberflächen effiziente Elektronenstrukturmethoden, die in der Lage sind sowohl kovalente also auch nichtkovalente Wechselwirkungen verlässlich zu beschreiben. In diesem Kontext stellen wir eine Methode im Bereich der Dichtefunk- tionalnäherungen (DFA) vor, die abgeschirmte vdW-Wechselwirkungen beinhaltet (DFA+vdWsurf), um die Adsorption von Atomen und Molekülen auf Oberflächen zu modellieren. Im Besonderen kombinieren wir dispersionskorrigierte Dichtefunktional- näherungen (die Tkatchenko-Scheffler DFA+vdW Methode) mit der Lifshitz-Zaremba- Kohn-Theorie um die Coulomb-Abschirmung innerhalb der Substratoberfläche bei der Berechnung der vdW-C6-Koeffizienten und vdW-Radien zu berücksichtigen. Unsere Methode inkludiert sowohl Polarisationseffekte der Bildebene und der Grenzfläche ab durch semilokale Hybridisierung aufgrund der Abhängigkeit der interatomaren C6 - Parameter von der Elektronendichte. Wir zeigen, dass die Berücksichtigung der semilokalen Mehrteilchenantwort der Substratelektronen essentiell für eine zuverläs- sige Vorhersage der Struktur und Stabilität der Atome und Moleküle auf der Oberfläche ist. Als Beispiele verwenden wir die Adsorption einer Xe-Monolage und eines aromatis- chen Moleküls, das Sauerstoffatome beinhaltet. Im Speziellen demonstrieren wir, dass die DFA+vdWsurf-Methode für die Adsorp- tion von 3,4,9,10-Perylentetracarbonsäuredianhydrid (C24H8O6, PTCDA) auf Ag(111), iii iv Au(111) und Cu(111) Geometrien liefert, die bemerkenswert gut (innerhalb von etwa 0.1 Å) mit normal incidence x-ray standing wave Messunge übereinstimmen. Weitere Beispiele beinhalten die Adsorption von Xe auf übergangsmetall-Oberflächen und eine vergleichende Studie der Grenzflächen, die von einer PTCDA-Monolage auf Ag(111), Ag(100) und Ag(110)-Oberflächen gebildet werden, um zu zeigen, dass diese Meth- ode Oberflächenempfindlichkeit aufweist. Diese Resultate demonstrieren, dass die DFA+vdWsurf-Methode verlässlich mit einer großen Bandbreite von Wechselwirkungen der HIOS (chemische Wechselwirkungen, elektrostatische Wechselwirkungen, Pauli- Abstoßung und vdW-Wechselwirkungen) umgehen kann. Daher stellt sie durch ihre Ef- fizienz und Leistbarkeit im Bezug auf Rechenzeit eine vernünftige Option für die präzise Beschreibung von Adsorptionsproblemen dar. Allerdings führen kooperative elektron- ische Effekte zwischen Atomen in größeren Molekülen zu einer Nicht-Additivität der molekularen Polarisierbarkeiten. Dieser Effekt wird in der DFA+vdWsurf Methode nicht berücksichtigt. Abschließend wird dies und andere verbleibende Herausforderungen für die Erzielung von quantitativer Genauigkeit und Vorhersagekraft bei der Simulation von der Struktur und Stabilität komplexer Grenzflächen diskutiert. v Acknowledgements I would first like to express my gratitude to my supervisor Alexandre Tkatchenko for his continuous support and invaluable contributions, specially during the last stages, of my doctoral research project. The way he solves complex problems using (relative) simple concepts is the reason for the success of this project. I would also like to thank Matthias Scheffler for giving me the opportunity of pursuing my Ph.D. studies at the Theory De- partment of the Fritz-Haber-Institut, letting me accomplish an important personal as- piration in my professional life. My sincere gratitude also goes to Wei Liu, whose con- tributions were key for the success of this project. I would also like to thank Guo-Xu Zhang, Oliver Hofmann, David Egger, and Egbert Zojer for their valuable collaboration and insights throughout the progress of this project. I would also like to take this opportunity to thank my teachers Rossitza Pentcheva, Nikolaj Moll,
Load more

Recommended publications
  • An Introduction to Topology the Classification Theorem for Surfaces by E
    An Introduction to Topology An Introduction to Topology The Classification theorem for Surfaces By E. C. Zeeman Introduction. The classification theorem is a beautiful example of geometric topology. Although it was discovered in the last century*, yet it manages to convey the spirit of present day research. The proof that we give here is elementary, and its is hoped more intuitive than that found in most textbooks, but in none the less rigorous. It is designed for readers who have never done any topology before. It is the sort of mathematics that could be taught in schools both to foster geometric intuition, and to counteract the present day alarming tendency to drop geometry. It is profound, and yet preserves a sense of fun. In Appendix 1 we explain how a deeper result can be proved if one has available the more sophisticated tools of analytic topology and algebraic topology. Examples. Before starting the theorem let us look at a few examples of surfaces. In any branch of mathematics it is always a good thing to start with examples, because they are the source of our intuition. All the following pictures are of surfaces in 3-dimensions. In example 1 by the word “sphere” we mean just the surface of the sphere, and not the inside. In fact in all the examples we mean just the surface and not the solid inside. 1. Sphere. 2. Torus (or inner tube). 3. Knotted torus. 4. Sphere with knotted torus bored through it. * Zeeman wrote this article in the mid-twentieth century. 1 An Introduction to Topology 5.
    [Show full text]
  • Surface Physics I and II
    Surface physics I and II Lectures: Mon 12-14 ,Wed 12-14 D116 Excercises: TBA Lecturer: Antti Kuronen, [email protected] Exercise assistant: Ane Lasa, [email protected] Course homepage: http://www.physics.helsinki.fi/courses/s/pintafysiikka/ Objectives ● To study properties of surfaces of solid materials. ● The relationship between the composition and morphology of the surface and its mechanical, chemical and electronic properties will be dealt with. ● Technologically important field of surface and thin film growth will also be covered. Surface physics I 2012: 1. Introduction 1 Surface physics I and II ● Course in two parts ● Surface physics I (SPI) (530202) ● Period III, 5 ECTS points ● Basics of surface physics ● Surface physics II (SPII) (530169) ● Period IV, 5 ECTS points ● 'Special' topics in surface science ● Surface and thin film growth ● Nanosystems ● Computational methods in surface science ● You can take only SPI or both SPI and SPII Surface physics I 2012: 1. Introduction 2 How to pass ● Both courses: ● Final exam 50% ● Exercises 50% ● Exercises ● Return by ● email to [email protected] or ● on paper to course box on the 2nd floor of Physicum ● Return by (TBA) Surface physics I 2012: 1. Introduction 3 Table of contents ● Surface physics I ● Introduction: What is a surface? Why is it important? Basic concepts. ● Surface structure: Thermodynamics of surfaces. Atomic and electronic structure. ● Experimental methods for surface characterization: Composition, morphology, electronic properties. ● Surface physics II ● Theoretical and computational methods in surface science: Analytical models, Monte Carlo and molecular dynamics sumilations. ● Surface growth: Adsorption, desorption, surface diffusion. ● Thin film growth: Homoepitaxy, heteroepitaxy, nanostructures.
    [Show full text]
  • Surface Water
    Chapter 5 SURFACE WATER Surface water originates mostly from rainfall and is a mixture of surface run-off and ground water. It includes larges rivers, ponds and lakes, and the small upland streams which may originate from springs and collect the run-off from the watersheds. The quantity of run-off depends upon a large number of factors, the most important of which are the amount and intensity of rainfall, the climate and vegetation and, also, the geological, geographi- cal, and topographical features of the area under consideration. It varies widely, from about 20 % in arid and sandy areas where the rainfall is scarce to more than 50% in rocky regions in which the annual rainfall is heavy. Of the remaining portion of the rainfall. some of the water percolates into the ground (see "Ground water", page 57), and the rest is lost by evaporation, transpiration and absorption. The quality of surface water is governed by its content of living organisms and by the amounts of mineral and organic matter which it may have picked up in the course of its formation. As rain falls through the atmo- sphere, it collects dust and absorbs oxygen and carbon dioxide from the air. While flowing over the ground, surface water collects silt and particles of organic matter, some of which will ultimately go into solution. It also picks up more carbon dioxide from the vegetation and micro-organisms and bacteria from the topsoil and from decaying matter. On inhabited watersheds, pollution may include faecal material and pathogenic organisms, as well as other human and industrial wastes which have not been properly disposed of.
    [Show full text]
  • 3D Modeling: Surfaces
    CS 430/536 Computer Graphics I Overview • 3D model representations 3D Modeling: • Mesh formats Surfaces • Bicubic surfaces • Bezier surfaces Week 8, Lecture 16 • Normals to surfaces David Breen, William Regli and Maxim Peysakhov • Direct surface rendering Geometric and Intelligent Computing Laboratory Department of Computer Science Drexel University 1 2 http://gicl.cs.drexel.edu 1994 Foley/VanDam/Finer/Huges/Phillips ICG 3D Modeling Representing 3D Objects • 3D Representations • Exact • Approximate – Wireframe models – Surface Models – Wireframe – Facet / Mesh – Solid Models – Parametric • Just surfaces – Meshes and Polygon soups – Voxel/Volume models Surface – Voxel – Decomposition-based – Solid Model • Volume info • Octrees, voxels • CSG • Modeling in 3D – Constructive Solid Geometry (CSG), • BRep Breps and feature-based • Implicit Solid Modeling 3 4 Negatives when Representing 3D Objects Representing 3D Objects • Exact • Approximate • Exact • Approximate – Complex data structures – Lossy – Precise model of – A discretization of – Expensive algorithms – Data structure sizes can object topology the 3D object – Wide variety of formats, get HUGE, if you want each with subtle nuances good fidelity – Mathematically – Use simple – Hard to acquire data – Easy to break (i.e. cracks represent all primitives to – Translation required for can appear) rendering – Not good for certain geometry model topology applications • Lots of interpolation and and geometry guess work 5 6 1 Positives when Exact Representations Representing 3D Objects • Exact
    [Show full text]
  • High-Throughput Investigation of the Geometry and Electronic Structures
    Subscriber access provided by FRITZ HABER INST DER MPI Article High-Throughput Investigation of the Geometry and Electronic Structures of Gas-Phase and Crystalline Polycyclic Aromatic Hydrocarbons Bohdan Schatschneider, Stephen Monaco, Jian-Jie Liang, and Alexandre Tkatchenko J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp5064462 • Publication Date (Web): 07 Aug 2014 Downloaded from http://pubs.acs.org on August 12, 2014 Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.
    [Show full text]
  • Surface Topology
    2 Surface topology 2.1 Classification of surfaces In this second introductory chapter, we change direction completely. We dis- cuss the topological classification of surfaces, and outline one approach to a proof. Our treatment here is almost entirely informal; we do not even define precisely what we mean by a ‘surface’. (Definitions will be found in the following chapter.) However, with the aid of some more sophisticated technical language, it not too hard to turn our informal account into a precise proof. The reasons for including this material here are, first, that it gives a counterweight to the previous chapter: the two together illustrate two themes—complex analysis and topology—which run through the study of Riemann surfaces. And, second, that we are able to introduce some more advanced ideas that will be taken up later in the book. The statement of the classification of closed surfaces is probably well known to many readers. We write down two families of surfaces g, h for integers g ≥ 0, h ≥ 1. 2 2 The surface 0 is the 2-sphere S . The surface 1 is the 2-torus T .For g ≥ 2, we define the surface g by taking the ‘connected sum’ of g copies of the torus. In general, if X and Y are (connected) surfaces, the connected sum XY is a surface constructed as follows (Figure 2.1). We choose small discs DX in X and DY in Y and cut them out to get a pair of ‘surfaces-with- boundaries’, coresponding to the circle boundaries of DX and DY.
    [Show full text]
  • Differential Geometry of Curves and Surfaces 3
    DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 3. Regular Surfaces 3.1. The definition of a regular surface. Examples. The notion of surface we are going to deal with in our course can be intuitively understood as the object obtained by a potter full of phantasy who takes several pieces of clay, flatten them on a table, then models of each of them arbitrarily strange looking pots, and finally glue them together and flatten the possible edges and corners. The resulting object is, formally speaking, a subset of the three dimensional space R3. Here is the rigorous definition of a surface. Definition 3.1.1. A subset S ⊂ R3 is a1 regular surface if for any point P in S one can find an open subspace U ⊂ R2 and a map ϕ : U → S of the form ϕ(u, v)=(x(u, v),y(u, v), z(u, v)), for (u, v) ∈ U with the following properties: 1. The function ϕ is differentiable, in the sense that the components x(u, v), y(u, v) and z(u, v) have partial derivatives of any order. 2 3 2. For any Q in U, the differential map d(ϕ)Q : R → R is (linear and) injective. 3. There exists an open subspace V ⊂ R3 which contains P such that ϕ(U) = V ∩ S and moreover, ϕ : U → V ∩ S is a homeomorphism2. The pair (U, ϕ) is called a local parametrization, or a chart, or a local coordinate system around P . Conditions 1 and 2 say that (U, ϕ) is a regular patch.
    [Show full text]
  • Mostly Surfaces
    Mostly Surfaces Richard Evan Schwartz ∗ August 21, 2011 Abstract This is an unformatted version of my book Mostly Surfaces, which is Volume 60 in the A.M.S. Student Library series. This book has the same content as the published version, but the arrangement of some of the text and equations here is not as nice, and there is no index. Preface This book is based on notes I wrote when teaching an undergraduate seminar on surfaces at Brown University in 2005. Each week I wrote up notes on a different topic. Basically, I told the students about many of the great things I have learned about surfaces over the years. I tried to do things in as direct a fashion as possible, favoring concrete results over a buildup of theory. Originally, I had written 14 chapters, but later I added 9 more chapters so as to make a more substantial book. Each chapter has its own set of exercises. The exercises are embedded within the text. Most of the exercises are fairly routine, and advance the arguments being developed, but I tried to put a few challenging problems in each batch. If you are willing to accept some results on faith, it should be possible for you to understand the material without working the exercises. However, you will get much more out of the book if you do the exercises. The central object in the book is a surface. I discuss surfaces from many points of view: as metric spaces, triangulated surfaces, hyperbolic surfaces, ∗ Supported by N.S.F.
    [Show full text]
  • Basics of the Differential Geometry of Surfaces
    Chapter 20 Basics of the Differential Geometry of Surfaces 20.1 Introduction The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We simply want to introduce the concepts needed to understand the notion of Gaussian curvature, mean curvature, principal curvatures, and geodesic lines. Almost all of the material presented in this chapter is based on lectures given by Eugenio Calabi in an upper undergraduate differential geometry course offered in the fall of 1994. Most of the topics covered in this course have been included, except a presentation of the global Gauss–Bonnet–Hopf theorem, some material on special coordinate systems, and Hilbert’s theorem on surfaces of constant negative curvature. What is a surface? A precise answer cannot really be given without introducing the concept of a manifold. An informal answer is to say that a surface is a set of points in R3 such that for every point p on the surface there is a small (perhaps very small) neighborhood U of p that is continuously deformable into a little flat open disk. Thus, a surface should really have some topology. Also,locally,unlessthe point p is “singular,” the surface looks like a plane. Properties of surfaces can be classified into local properties and global prop- erties.Intheolderliterature,thestudyoflocalpropertieswascalled geometry in the small,andthestudyofglobalpropertieswascalledgeometry in the large.Lo- cal properties are the properties that hold in a small neighborhood of a point on a surface. Curvature is a local property. Local properties canbestudiedmoreconve- niently by assuming that the surface is parametrized locally.
    [Show full text]
  • Surface Details
    Surface Details ✫ Incorp orate ne details in the scene. ✫ Mo deling with p olygons is impractical. ✫ Map an image texture/pattern on the surface Catmull, 1974; Blin & Newell, 1976. ✫ Texture map Mo dels patterns, rough surfaces, 3D e ects. ✫ Solid textures 3D textures to mo del wood grain, stains, marble, etc. ✫ Bump mapping Displace normals to create shading e ects. ✫ Environment mapping Re ections of environment on shiny surfaces. ✫ Displacement mapping Perturb the p osition of some pixels. CPS124, 296: Computer Graphics Surface Details Page 1 Texture Maps ✫ Maps an image on a surface. ✫ Each element is called texel. ✫ Textures are xed patterns, pro cedurally gen- erated, or digitized images. CPS124, 296: Computer Graphics Surface Details Page 2 Texture Maps ✫ Texture map has its own co ordinate system; st-co ordinate system. ✫ Surface has its own co ordinate system; uv -co ordinates. ✫ Pixels are referenced in the window co ordinate system Cartesian co ordinates. Texture Space Object Space Image Space (s,t) (u,v) (x,y) CPS124, 296: Computer Graphics Surface Details Page 3 Texture Maps v u y xs t s x ys z y xs t s x ys z CPS124, 296: Computer Graphics Surface Details Page 4 Texture Mapping CPS124, 296: Computer Graphics Surface Details Page 5 Texture Mapping Forward mapping: Texture scanning ✫ Map texture pattern to the ob ject space. u = f s; t = a s + b t + c; u u u v = f s; t = a s + b t + c: v v v ✫ Map ob ject space to window co ordinate sys- tem. Use mo delview/pro jection transformations.
    [Show full text]
  • AN INTRODUCTION to the CURVATURE of SURFACES by PHILIP ANTHONY BARILE a Thesis Submitted to the Graduate School-Camden Rutgers
    AN INTRODUCTION TO THE CURVATURE OF SURFACES By PHILIP ANTHONY BARILE A thesis submitted to the Graduate School-Camden Rutgers, The State University Of New Jersey in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Mathematics written under the direction of Haydee Herrera and approved by Camden, NJ January 2009 ABSTRACT OF THE THESIS An Introduction to the Curvature of Surfaces by PHILIP ANTHONY BARILE Thesis Director: Haydee Herrera Curvature is fundamental to the study of differential geometry. It describes different geometrical and topological properties of a surface in R3. Two types of curvature are discussed in this paper: intrinsic and extrinsic. Numerous examples are given which motivate definitions, properties and theorems concerning curvature. ii 1 1 Introduction For surfaces in R3, there are several different ways to measure curvature. Some curvature, like normal curvature, has the property such that it depends on how we embed the surface in R3. Normal curvature is extrinsic; that is, it could not be measured by being on the surface. On the other hand, another measurement of curvature, namely Gauss curvature, does not depend on how we embed the surface in R3. Gauss curvature is intrinsic; that is, it can be measured from on the surface. In order to engage in a discussion about curvature of surfaces, we must introduce some important concepts such as regular surfaces, the tangent plane, the first and second fundamental form, and the Gauss Map. Sections 2,3 and 4 introduce these preliminaries, however, their importance should not be understated as they lay the groundwork for more subtle and advanced topics in differential geometry.
    [Show full text]
  • Chemical Stability and Performance Influence of Choice Substituents and Core Conjugation of Organic Semiconductors
    University of Massachusetts Amherst ScholarWorks@UMass Amherst Doctoral Dissertations Dissertations and Theses March 2019 CHEMICAL STABILITY AND PERFORMANCE INFLUENCE OF CHOICE SUBSTITUENTS AND CORE CONJUGATION OF ORGANIC SEMICONDUCTORS Jack Ly University of Massachusetts Amherst Follow this and additional works at: https://scholarworks.umass.edu/dissertations_2 Part of the Polymer and Organic Materials Commons, Polymer Science Commons, and the Semiconductor and Optical Materials Commons Recommended Citation Ly, Jack, "CHEMICAL STABILITY AND PERFORMANCE INFLUENCE OF CHOICE SUBSTITUENTS AND CORE CONJUGATION OF ORGANIC SEMICONDUCTORS" (2019). Doctoral Dissertations. 1516. https://doi.org/10.7275/13323135 https://scholarworks.umass.edu/dissertations_2/1516 This Open Access Dissertation is brought to you for free and open access by the Dissertations and Theses at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact [email protected]. CHEMICAL STABILITY AND PERFORMANCE INFLUENCE OF CHOICE SUBSTITUENTS AND CORE CONJUGATION OF ORGANIC SEMICONDUCTORS A Dissertation Presented By JACK THANH LY Submitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY February 2019 Polymer Science and Engineering © Copyright by Jack Thanh Ly 2019 All Rights Reserved CHEMICAL STABILITY AND PERFORMANCE INFLUENCE OF CHOICE
    [Show full text]