Multiscale Materials Modelling

Fundamentals and Applications

Z. X. Guo

• Publisher: Woodhead Publishing
• Year: 2007
• Print ISBN: 9781845690717
• eBook ISBN: 9781845693374
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

Multiscale materials modelling offers an integrated approach to modelling material behaviour across a range of scales from the electronic, atomic and microstructural up to the component level. As a result, it provides valuable new insights into complex structures and their properties, opening the way to develop new, multi-functional materials together with improved process and product designs. Multiscale materials modelling summarises some of the key techniques and their applications.The various chapters cover the spectrum of scales in modelling methodologies, including electronic structure calculations, mesoscale and continuum modelling. The book covers such themes as dislocation behaviour and plasticity as well as the modelling of structural materials such as metals, polymers and ceramics. With its distinguished editor and international team of contributors, Multiscale materials modelling is a valuable reference for both the modelling community and those in industry wanting to know more about how multiscale materials modelling can help optimise product and process design. - Reviews the principles and applications of mult-scale materials modelling - Covers themes such as dislocation behaviour and plasticity and the modelling of structural materials - Examines the spectrum of scales in modelling methodologies, including electronic structure calculations, mesoscale and continuum modelling

1

The role of ab initio electronic structure calculations in multiscale modelling of materials

M. Šob Masaryk University, Czech Republic and Academy of Sciences of the Czech Republic

1.1 Introduction

Most, if not all, of the properties of solids can be traced to the behavior of electrons, the ‘glue’ that holds atoms together to form a solid. An important aim of the condensed matter theory is thus calculating the electronic structure (ES) of solids. The theory of ES is not ony helpful in understanding and interpreting experiments, but it also becomes a predictive tool of the physics and chemistry of condensed matter and materials science.

Many of the structural and dynamical properties of solids can be predicted accurately from ab initio (first-principles) electronic structure calculations, i.e. from the fundamental quantum theory. Here the atomic numbers of constituent atoms and some structural information are employed as the only pieces of input data. Such calculations are routinely performed within the framework of density functional theory in which the complicated many-body motion of all electrons is replaced by an equivalent but simpler problem of a single electron moving in an effective potential.

A general formulation of the quantum mechanical equations for ES including all known interactions between the electrons and atomic nuclei in solids is relatively simple, but we are still not able to solve these equations in their full generality. A great many approximations must be performed, which, in many cases, leads to a comprehensive solution. Its analysis brings us then some understanding of various phenomena and processes in condensed matter. The ES problem is computationally very demanding. This is why practical ES calculations in solids were rather rare prior to the availability of larger high-speed computers.

Since the 1980s, ES theory has exhibited a growing ability to understand and predict material properties and to use computers to design new materials. A new field of solid-state physics and materials science has emerged – computational materials science. This has achieved a considerable degree of reliability concerning predictions of physical and chemical properties and phenomena, thanks in large part to continued rapid development and availability of computing power (speed and memory), its increasing accessibility (via networks and workstations), and to the generation of new computational methods and algorithms which this enabled. State-of-the-art ES calculations yield highly precise solutions to the one-electron Kohn–Sham equation for a solid and provide an understanding of matter at the atomic and electronic scale with an unprecedented level of detail and accuracy. In many cases, we are able not only to simulate experiment but also to design new molecules and materials and to predict their properties before actually synthesizing them. A computational simulation can also provide data on the atomic scale that are inaccessible experimentally. In contrast to semi-empirical approaches – the adjustable parameters of which are fitted to the properties of the ground state structure and, therefore, may not be transferable to non-equilibrium configurations – ab initio calculations are reliable far from the equilibrium as well.

In multiscale modelling of materials, the role of ab initio electronic structure calculations is twofold: (i) to study the situations where the electronic effects are crucial and must be treated from first principles and (ii) to provide data for generation of interatomic potentials. In this chapter, we will discuss both these aspects. Let us note that there exists a vast literature devoted to multiscale modelling of materials. Recent reviews may be found for example in^1,2 and in the Handbook of Materials Modeling edited by S. Yip³, the latest developments are documented in the proceedings of various meetings on multiscale modelling of materials (the latest one took place in September 2006 in Freiburg, Germany⁴).

1.2 Basic equations of electronic structure calculations

In a solid where relativistic effects are not essential, we may describe the electron states by the non-relativistic many-electron Schrödinger equation

${\mathcal{H}}_{\mathrm{e},\left\{{\mathbf{R}}_{\mathbf{\alpha }}\right\}}\mathrm{\Psi }=\mathit{E}\mathrm{\Psi }$

[1.1]

with the Hamiltonian

${\mathcal{H}}_{\mathrm{e},\left\{{\mathbf{R}}_{\mathit{\alpha }}\right\}}=-{\displaystyle \sum _{\mathit{i}}{\nabla }_{\mathit{i}}^2}+{\displaystyle \sum _{\mathit{i}}{\mathit{V}}_{\mathrm{e},\left\{{\mathbf{R}}_{\mathit{\alpha }}\right\}}}\left({\mathbf{r}}_{\mathit{i}}\right)+{\displaystyle \sum _{\mathit{i},\mathit{j}}\prime }\frac{1}{\left|{\mathbf{r}}_{\mathit{i}}-{\mathbf{r}}_{\mathit{j}}\right|},$

[1.2]

where {R_α} are the instantaneous positions of the atomic nuclei, {r_i} denote positions of electrons, the ${\mathit{V}}_{\mathrm{e},\left\{{\mathbf{R}}_{\mathit{\alpha }}\right\}}\left({\mathbf{r}}_{\mathit{i}}\right)$

is the potential experienced by the ith electron in the field of all nuclei at the positions {R_α} with the atomic numbers Z_α, i.e.

${\mathit{V}}_{\mathrm{e},\left\{{\mathbf{R}}_{\mathit{\alpha }}\right\}}\left({\mathbf{r}}_{\mathit{i}}\right)=-{\displaystyle \sum _{\mathit{\alpha }}\frac{2{\mathit{Z}}_{\mathit{\alpha }}}{\left|{\mathbf{r}}_{\mathit{i}}-{\mathbf{R}}_{\mathit{\alpha }}\right|}}$

[1.3]

and the last term in equation [1.2] represents the electrostatic electron–electron interaction (the prime on the summation excludes i = j). Let us note that here and throughout the chapter we use Rydberg atomic units with ħ = 1, 2m_e = 1 and e² = 2, where m_e and e denote the electron mass and charge, ...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright page
• Contributor contact details
• Preface
• 1: The role of ab initio electronic structure calculations in multiscale modelling of materials
• 2: Modelling of dislocation behaviour at the continuum level
• 3: Phase-field modelling of material microstructure
• 4: Mesoscale modelling of grain growth and microstructure in polycrystalline materials
• 5: Finite element and homogenization modelling of materials
• 6: Grain–continuum modelling of material behaviour
• 7: Coupled atomistic/continuum modelling of plasticity in materials
• 8: Multiscale modelling of carbon nanostructures
• 9: Multiscale modelling of structural materials
• Index