SPRINGER BRIEFS IN APPLIED SCIENCES AND TECHNOLOGY Wolfgang Gräfe Quantum Mechanical Models of Metal Surfaces and Nanoparticles 123 SpringerBriefs in Applied Sciences and Technology www.com More information about this series at http://www.com/series/8884 www.com Wolfgang Gräfe Quantum Mechanical Models of Metal Surfaces and Nanoparticles 123 www.com Wolfgang Gräfe Berlin Germany ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISBN 978-3-319-19763-0 ISBN 978-3-319-19764-7 (eBook) DOI 10.1007/978-3-319-19764-7 Library of Congress Control Number: 2015941358 Springer Cham Heidelberg New York Dordrecht London © The Author(s) 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.com Preface In this book I consider two simple quantum mechanical models of metal surfaces. It is the aim to give an ostensive picture of the forces acting in a metal surface and to deduce analytical formulae for the description of their physical properties.
The starting points of my approach to the surface physics were strength and fatigue limit. As the cause of these features I consider a near-surface stress with the dimension of a force per area. In this book I explain the relation between the near-surface stress and the familiar surface parameters. In order to make the understanding of my theory easier I have applied the concept of the separation of the three-dimensional body into three one-dimensional subsystems.
This book has been written for experts and newcomers in the field of surface physics. Wolfgang Gräfe v www.com Acknowledgments Without the patience and without the care of my wife Herta I would not have accomplished this book.com Contents 1 Introduction .1 Electrocapillarity of Liquids .2 Surface Free Energy and Surface Stress of Solids .3 The Estance or the Surface Stress-Charge Coefficient .4 Experimental Data in the Literature .5 State of the Theoretical Knowledge.6 The Aim of the Following Text. 6 2 The Model of Kronig and Penney .1 The Density of the Electron Energy Levels n(E). 13 3 Tamm’s Electronic Surface States.
18 4 The Extension of the Kronig–Penney Model by Binding Forces. 19 5 The Separation of the Semi-infinite Model and the Calculation of the Surface Parameters for the Three-Dimensional body at T = 0 K (Regula Falsi of Surface Theory) .1 The Separability of the Chemical Potential .2 The Separability of the Fermi Distribution Function .3 The Calculation of Surface Energy, Surface Stress, and Surface Charge at T = 0 K (Regula Falsi of Surface Theory) .com x Contents 6 The Surface Parameters for the Semi-infinite Three-Dimensional Body at Arbitrary Temperature. 41 7 The Surface Free Energy φ and the Point of Zero Charge Determined for the Semi-infinite Model .1 Electron Transitions from the Bulk into the Surface and the Contribution to the Surface Free Energy φTr .2 The Point of Zero Charge (PZC) and the Fermi Level Shift .3 The Contribution of the Electrostatic Repulsion Between the Electrons in the Surface Bands to the Surface Energy. 51 8 A Model with a Limited Number of Potential Wells .1 Modeling a Nanoparticle and a Solid Surface .2 The Energy of the Electrons in the Bulk and in the Surface Bands .3 Calculation of the Surface Parameters .1 The Surface Energy uESB of the Electrons in a Surface Band of a Nanocube .2 Calculation of the Surface Free Energy u in a Nanocube.3 Calculation of the Surface Stress for a Nanocube and a Plate-like Body .5 The Surface Charge Densities and the Point of Zero Charge in a Nanocube.
68 9 Surface Stress-Charge Coefficient (Estance). 69 10 Regard to the Spin in the Foregoing Texts. 73 11 Detailed Calculation of the Convolution Integrals. 80 12 Comparison of the Results for the Semi-infinite and the Limited Body .1 The Semi-infinite Body .2 Density Distribution of the Energy Levels .4 Surface Free Energy .com Contents xi 12.2 The Limited Body.2 Density Distribution of the Energy Levels .3 Surface Free Energy of a Nanocube .4 Surface Free Energy of a Plate-like Body.
84 13 Calculation of Surface Stress and Herring’s Formula. 89 14 Miscellaneous and Open Questions .1 The Scientific Ambition of this Book .1 Semi-infinitely Extended Body at 300 K .2 Nanocube of 10 × 10 × 10 Potential Wells at 0 K .3 Support for the Presented Theory .5 Surface Stress and Young’s Modulus .7 The Minimum of the Surface Free Energy .8 Fermi Level/Chemcal Potential .9 The Influence of the Number of Atoms on the Results .com Nomenclature a Width of potential well b Width of potential barrier c Lattice constant A Surface area de Electron density e Absolute value of electron charge E Energy EB Bottom of the energy band EF Fermi-level ES Energy of the surface state ESa Energy of an additional surface state ESo Energy of an offspring surface state ET Top of the energy band Eul Energy level in an unlimited (infinitely extended) body ESB Electrons in the surface band (superscript) fi ith Component of the force per electron F Force FSi ith Component of the force in the surface layer ħ Reduced Planck’s constant i Imaginary unit i Index j Index k Index k Boltzmann’s constant k Wave number L Length of an edge m Electron mass n Density of electrons N Number of electrons NA Number of an atom Nel Total number of energy levels xiii www.com xiv Nomenclature p Fermi distribution function, probability pl Plate-like (superscript) q Surface charge density qESB Surface charge resulting from the electrons in the surface band r Radius R Radius sij Near-surface stress S Surface state (superscript) Sa Additional surface state (superscript) So Offspring surface state (superscript) T Absolute temperature TR Transition (superscript) ul Unlimited (superscript) U Potential barrier in the bulk US Potential barrier at the surface x Coordinate X Number of mols Y Young’s modulus, modulus of elasticity Z Number of particles Z Partition function γ Surface tension δ Attenuation length, penetration depth of a wave function Δ Laplace operator δ(x) Delta function ε Dielectric coefficient ε Strain εij Strain tensor ε0 Absolute permittivity ϕ Potential difference across the interface φ Surface free energy φESB Surface energy resulting from the electrons in the surface band φTR Contribution to the surface free energy due to electron transition μ Index ν Index ψ Wave function ςij Estance, surface stress-charge coefficient σij Surface stress tensor σESB ij Surface stress tensor resulting from the electrons in the surface band ζ Chemical potential Ω Volume of the phase space www.com Chapter 1 Introduction Abstract The surface free energy is the work performed from outside for the generation of an additional surface. The surface tension is the force per length acting on the surface. For a liquid the amount of the surface free energy per unit area equals the surface tension.
Electrocapillarity means the change of the surface tension due to the influence of a surface charge. The surface tension reaches a maximum if the surface charge density vanishes. For a solid the quantities “surface free energy” and “surface tension” are different. Shuttleworth has formulated a relation between the surface free energy per unit area φ and the surface stress σ du r¼uþA : dA T A compilation of experimental and theoretical data in the literature is given for the surface parameters of solids.
It is the aim of the book to classify the notion “near-surface stress” in the list of surface quantities. The near-surface stress has been used by the author for an explanation of the cause of fatigue limit in strength investigations. Keywords Surface energy of solids Surface stress of solids The surface free energy is the work performed from outside for the generation of an additional surface. The surface tension is the force per length acting on the surface.
For a liquid the amount of the surface free energy per unit area equals the surface tension. As a matter of principle, for a liquid the experimental determination of the surface tension is a straightforward procedure.1 Electrocapillarity of Liquids In the case of liquids, electrocapillarity means the change of the surface tension due to the influence of a surface charge. Gräfe, Quantum Mechanical Models of Metal Surfaces and Nanoparticles, SpringerBriefs in Applied Sciences and Technology, DOI 10.com 2 1 Introduction If the difference of the electric potential across the interface between a mercury electrode and a surrounding electrolyte is varied, the surface tension γ of the mercury changes too. For a certain potential difference ϕ across the interface, the surface charge density q in the mercury is zero.
According to the Lippmann–Helmholtz equation @c ¼ qM ¼ qL ð1:1Þ @/ p;T;f the surface charge density is referred to the surface tension γ. The surface tension reaches a maximum if the surface charge density vanishes.) The symbol ζ means the chemical potential, qM and qL are the excess charge densities in the metal and in the electrolyte at the phase interface. In the experimental situation considered here, the surface tension γ is the Gibbs surface free energy per surface area. Due to the mutual repulsion of the charges in the surface layer a negative contribution to the surface energy and the surface stress will arise which is zero for a vanishing surface charge density q.2 Surface Free Energy and Surface Stress of Solids As Gibbs [2] has pointed out, for a solid the quantities “surface free energy” and “surface tension” are different at least in their nature.
Here too, the surface free energy is the work performed from outside in the production of an additional surface. Furthermore, we have to consider stresses and strains in the solid surface by which no additional atom will be introduced into the surface. It is always possible to find two perpendicular directions in the surface for which no shear stresses exist. The surface stress components in these two directions are the prin- cipal surface stresses.
For the case of a solid, Shuttleworth [3] has defined the surface tension as the arithmetic mean of the values of the principal surface stresses. According to him (citation), “for an isotropic substance, or for a crystal face with a three- (or greater-) fold axis of symmetry, all normal components of the surface stress equal the surface tension.” Shuttleworth [3] has formulated the following thermodynamic relation between the quantities, surface free energy per unit area φ and surface tension σ. du r¼uþA : ð1:2Þ dA T According to this relation, the surface tension σ is the sum of the surface free energy per unit area φ and its strain derivative. The letter A means the surface area.
For the surface stress tensor σij with the dimension of a force per length, Herring [4] deduced the formula www.2 Surface Free Energy and Surface Stress of Solids 3 @u rij ¼ udij þ : ð1:3Þ @eij Here δij is the Kronecker symbol and εij is the surface elastic strain tensor. In both second rank tensors εij and σij the indices i and j take only two values, e.