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Fundamentals of Stress and Strain at the Nanoscale Level: Towards Nanoelasticity Pierre Müller

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Fundamentals of Stress and Strain at the Nanoscale Level: Towards Nanoelasticity Pierre Müller Fundamentals of stress and strain at the nanoscale level: towards nanoelasticity Pierre Müller To cite this version: Pierre Müller. Fundamentals of stress and strain at the nanoscale level: towards nanoelasticity. Mechanical stress on the nanoscale, 2011. hal-01967052 HAL Id: hal-01967052 https://hal.archives-ouvertes.fr/hal-01967052 Submitted on 3 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Druckfreigabe/approval for printing Without corrections/ ` ohne Korrekturen j27 After corrections/ 1 nach Ausfçhrung ` 2 der Korrekturen 3 Date/Datum: ................................... 4 Signature/Zeichen: ............................ 5 6 7 8 2 9 Fundamentals of Stress and Strain at the Nanoscale Level: 10 Toward Nanoelasticity 11 € 12 Pierre Muller 13 14 2.1 15 Introduction 16 17 Understanding the relationship between the structure and the shape of a piece of 18 matter and its mechanical properties has always been one of the primary goals of 19 material science. Over the past decades, the rapid progress in the development of new 20 materials with a size of a few nanometers has opened a new field of scientific and 21 technological interest. From a fundamental viewpoint, one of the key features of 22 nanosized material is their high surface/bulk ratio with the consequence that 23 materials in small dimensions behave differently from their bulk counterpart. From 24 a technological viewpoint, nanomaterials are promising building blocks in future 25 devices, but the stresses they develop can be detrimental to their reliability or can be 26 used to modify their physical properties. Thus, for both fundamental and techno- 27 logical applications, there is a growing interest in the study of the mechanics of 28 nanosized objects. 29 For instance, it has been found that the elastic modulus, which measures the 30 proportionality between the applied stress and the measured strain, is size dependent 31 for dimension smaller than roughly 200 nm. However, according to the authors, 32 calculated or measured elastic moduli have been reported to increase or decrease with 33 the size of the object. From an experimental viewpoint, this discrepancy may come 34 from the difficulty of manipulation and the lack of characterization of such small 35 objects, as well as from the uncertainty of the complex boundary conditions between a 36 nanosized object and the micronic-sized tool used to measure its elastic properties. 37 From a theoretical viewpoint, there are still some difficulties to perfectly describe 38 elasticity at the nanoscale. Most of the models used for describing the mechanical 39 properties of a body are based on the classical linear and infinitesimal theory of 40 elasticity of continuum mediums [1, 2]. In such models, a macroscopic object is 41 considered as a continuum medium, the surface of which is a simple boundary at 42 which external forces are applied. It means that the classical macroscopic theory 43 ignores surface effects since it does not attribute any specific elastic properties to the 44 surface. However, because of the missing bonds, there is a redistribution of 45 Mechanical Stress on the Nanoscale: Simulation, Material Systems and Characterization Techniques, First Edition. Edited by Margrit Hanbucken,€ Pierre Muller,€ and Ralf B. Wehrspohn. Ó 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA. 28j 2 Fundamentals of Stress and Strain at the Nanoscale Level: Toward Nanoelasticity 1 electronic charges close to the surface that modifies the local binding properties so 2 that the surface layers cannot have the same elastic properties as the underlying bulk. 3 This means that because of the reduced coordination of surface atoms, a small piece 4 of matter cut in a homogeneous body is not homogeneous. The simpler way to 5 describe this inhomogeneity is to divide the object into a bulk core and a surface zone 6 with different elastic properties. We will show that when such specific surface 7 properties are considered, the elastic behavior of a finite-size body results from a 8 subtle interplay between its bulk and surface properties and thus depends on the 9 absolute size of the object. 10 Our goal in this chapter is to describe the elastic properties of nanoscale objects 11 from the point of view of surface physicists. In other words, our purpose is to properly 12 1) introduce surface effects on elastic classical theory. This chapter is divided into two 13 parts. The first part emphasizes the theoretical background. It begins by a few results 14 valid for the bulk (Section 2.2.1), followed by a description of the surface elastic 15 properties (Section 2.2.2) in the Gibbs meaning (Section 2.2.3) and then in the two- 16 phase models (Section 2.2.4). The second part concerns applications. We will 17 describe how surface stress may induce spontaneous deformation of nanoparticles 18 (Section 2.3.1), may modify the effective modulus of freestanding thin films 19 (Section 2.3.2), and may play a role for the static bending of thin films (Section 2.3.3) 20 and nanowires (Section 2.3.4). A short conclusion will sum up the main effects the 21 surfaces induce on the elastic properties of nanoscale objects. 22 Again, our goal is not to review all recent results, since in particular the number of 23 papers devoted to the study of the mechanical properties of nanosized objects is 24 increasing exponentially, but essentially to underline the fundamentals of 25 nanoelasticity. 26 27 28 2.2 29 Theoretical Background 30 31 2.2.1 32 Bulk Elasticity: A Recall 33 34 The classical theory of elasticity treats solids as continuum mediums in which the 35 deformation is described by continuum fields such as the displacement vector ~uð~rÞ 36 and the strain tensor eðÞ~r , while internal forces are described by the stress tensor sðÞ~r . 37 Within linear infinitesimal elasticity, the equilibrium deformation field is a solution 38 of a second-order equation that has to be solved with boundary equations expressed in 39 terms of external forces applied at the free surface of the body. 40 In this section, we will recall some classical results valid for bulk elasticity but 41 necessary as a prerequisite before studying surface elasticity in Section 2.2.2. 42 43 44 1) The limit of the description is that it is necessary to be able to discriminate surface from bulk. Such models thus cannot be used to describe the elastic properties of aggregates formed only by a few 45 atoms, for which it is no more possible to attribute to the atoms a surface or a bulk character. 2.2 Theoretical Background j29 1 2.2.1.1 Stress and Strain Definition 2 If a continuum medium is strained, a point in the body is displaced from its initial 3 ~ ~0 ¼~þ~ ~ ¼ 0À position r to r r u. The vector u pofffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi components ui xi xi is called the 4 ‘ ¼ 2 þ 2 þ 2 displacement vector. The distance d dx1 dx2 dx3 betweenpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi two points of 5 fi ~ ‘0 ¼ 02 þ 02 þ 02 in nitesimal vicinity dx before deformation becomes d dx1 dx2 dx3 6 0 ¼ þ with dxi dxi dui after deformation.P For small displacements, this distance can 7 ‘02 ¼ ‘2 þ be written as d d 2 i; jeijdxidxj, where the quantity 8 @ 9 1 @ui uj eij ¼ þ ð2:1Þ 10 2 @xj @xi 11 is called the bulk strain tensor. 12 When the body is strained, there develop forces tending to restore its initial 13 configuration. These internal forces are transmitted across the surface that bounds 14 ~ the volume. The forces F dS exerted on the faces of an elemental cube centered on the 15 considered point can be written for a face i as [1, 2] 16 X 17 Fi ¼ sijnj ð2:2Þ 18 j 19 ~ 20 where sij are the components of the so-called stress tensor and n is a unit vector 21 directed toward the exterior of the face i (see Figure 2.1). fi 22 Notice that in the framework of in nitesimal elasticity, the stress and strain fi 23 de nitions do not make a distinction between the deformed and the nondeformed fi 24 con guration of the body. This is not the case for large deformation [1, 2]. 25 26 2.2.1.2 Equilibrium State ¼ ; ; 27 From Eq. (2.2), theH forceP component Fi (i 1 2 3) on a volume V bounded by a ¼ 28 surface S is Fi jsijnjdS. It can be transformed into a volume integral 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 Figure 2.1 Action of the components sij of the sij. The first index i ¼ 1; 2; 3 gives the direction 44 bulk stress tensor applied on the three front xi where the stress acts. On the back faces of the faces of an elementary cube cut in a piece of cube, there are identical stresses of opposite 45 matter. Each face normal to xj axis bears a triplet sign. 30j 2 Fundamentals of Stress and Strain at the Nanoscale Level: Toward Nanoelasticity Ð P 1 ¼ ð@ =@ Þ Fi Ð j sij xj dV, and so taking into account the Newtons second law 2 2 2 Fi ¼ rðd ui=dt ÞdV (valid for a body of density r), we get 3 X 2 4 @s d u ij ¼ r i ð2:3Þ 5 @ 2 j xj dt 6 7 which is called the elastodynamic equation.
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