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**Chapter 3 Structure-Dependent Grain Boundary Properties**

**Introduction**

This chapter describes the initial stage of the multiscale computational study, entailing an assessment of fundamental structure-property relationships of grain boundaries using molecular dynamics simulations (hereafter MD).

Understanding the ‘physical causes’ of the structure-property relationships at the atomistic scale has been the subject of considerable interest since the latter 1970s [139, 140]. Although the relationship between GB free volume and potential energy is fairly well-established [141, 142], there remains a limited understanding about the influence of defects which can result in a highly non-equilibrium GB structural state [64, 143, 144]. Perhaps this is resultant from the overwhelming trend in atomistic studies; which emphasise the results of thermodynamically stable GBs. For this reason, the vast majority of atomistic studies of GBs in the literature are predicated on the assumption that a single GB structure is sufficient to model the characteristics at any misorientation angle and inclination plane [105, 107, 145, 146]. Such studies are useful for evaluating fundamental and ideal elastic properties of polycrystals, but not necessarily the non-equilibrium effects. This causes significant challenges when attempting to model the multi-variable structure-property-energy relationships. Despite this, a recent extensive review concluded that studies of GB energy show there is inherent variability and that future studies “must take account of the atomic structure and details of the bonding” [141].

Sutton et al. [147], hypothesized that GB structures with a high number of metastable states would exhibit lower energy barriers for defect interactions, due to accommodating mechanisms with GB structure transformation. The thermally-enhanced activation of defect nucleation from unstable GBs is inhibited by the activation energy barrier [148]. Correspondingly, the activation energy is evaluated directly from the difference between the current GB energy and that of the stable GB structure. Historically, a rigorous analysis of atomic effects has been limited by the lack of high quality, accurate inter-atomic potentials and insufficient computational processing power. Substantial developments in the NEB technique have also only become accessible in the latter part of this decade. Hence, a systematic analysis of the metastable GB structure equilibration mechanism by vacancy emission hypothesis (from [147]) has not been achieved until the present study. The interactions between vacancies and GBs have been implicated (and

recently measured [149]) to play a critical role during GB growth, migration and Coble creep, to facilitate diffusion effects and free volume shuffling [150]. Dislocation and vacancy interactions in real materials and/or thermal effects may cause non-equilibrium GB structures, in which case these metastable structures may play a role [151]. Furthermore, impurity atoms have been implicated in the process of GB structural ‘phases transformations’, causing multiplicity of GB structures in real materials [152]. In fact, the generation and emission of vacancies has been implicated as a geometrically necessary mechanism to enable GB migration and shrinking [153]. This occurs in order to release excess stored free volume that could not be eliminated elastically. These results also indicate that there is a direct relationship between the GB structure and the capacity for vacancy emission.

This chapter presents an investigation of the significance of GB structure, in terms of the multiplicity of available metastable or non-equilibrium bicrystals with MD and NEB simulations. Section 3.2 and Section 3.3 describe features of the common simulation set-up, and the approach to perform GB structural characterisation, respectively. A study of the mechanisms and origins of metastability in ‘realistic conditions’ is presented in Section 3.4. Section 3.5 discusses the implications of multiplicity of GB structure, and considers variability as a parameter that influences the interfacial susceptibility for dislocation interactions.

**Methods and Theory**

This chapter section describes the techniques and computational tools which were used for structure characterisation in subsequent chapter sections. The background theory and insights obtained from the successes and failures of the past are also provided to help affirm the validity of the approach described. It is noteworthy that in subsequent sub-sections, the procedure applied varies, depending on the application and characteristic of interest in the study, however the basic set-up to generate and equilibrate the bicrystals remains consistent.

**Classifying GB structures**

Any GB structure may be defined by five degrees of freedom, with three required to describe the relative tilt angle and axis for the rotation between the two grains, and two required to define the orientation of the boundary plane in either grain. A generalised approach to define a GB involves the following three parameters: (1) Misorientation angle, θ; (2) Misorientation tilt axis [ ]; and (3) Normal vector to the GB plane, N1,2= [ℎ ]. However for a symmetric tilt boundary, the boundary normals and tilt angles are complimentary.

**Bicrystal geometry**

For the atomistic analysis of GB properties in this chapter, a consistent bicrystal geometry was utilised in all cases with approximately identical sizes, however sometimes involved different interatomic potentials. This geometry was selected to eliminate independent variables associated with GB curvature, geometry-induced stress concentrations and to isolate a single GB structure. Furthermore, the utilisation of symmetric tilt GB structures eliminates any crystallographic discontinuity, as both single crystals have complimentary slip systems

The bicrystal construction was defined in such a manner that it resulted in the minimal simulation cell height that retained a nominal inter-GB spacing of 15 nm, which is consistent with previous studies [105, 111, 112]. Because we use MD simulations with 3D periodic boundary conditions, the specimen has two ‘quasi-infinite’ planar GBs at either interface between the two vertically stacked ‘quasi-infinite’ grains of 15 nm height. Figure 3.1a and 4b illustrate the characteristic bicrystal geometry utilised in many GB structural models in several chapters within this thesis.

The default crystallography utilised in the present study, is consistent with the description of a symmetric tilt Σ5(310) GB around the [001] tilt axis, with the crystallographic orientations illustrated in Figure 2.1a. However, a Σ13(510) bicrystal was also utilised in this chapter, subsection 3.5, and in Chapter 4, Section 4.4. The structure of a minimum energy pure-tilt GB is comprised of an array of ‘GB units’, such as shown in Figure 3.1c, which can be used to provide a simple but effective means of qualitative classification [140].

**Introduction**

1.1 Motivations

1.2 Scope of the Work

1.3 Thesis Outline

**2. Chapter 2 Literature Review**

2.1 Introduction

2.2 Polycrystal Plasticity

2.3 Grain Boundary Engineering

2.4 Molecular Dynamics

2.5 Nudged Elastic Band Method

2.6 Discrete Dislocation Dynamics

2.7 Multiscale Simulation Approaches

2.8 Prior Implementations of Polycrystal DD

2.9 Summary

**3. Chapter 3 Structure-Dependent Grain Boundary Properties**

3.1 Introduction

3.2 Methods and Theory

3.3 Structural Characterisation and GB Multiplicity

3.4 Defect Interactions and Thermal Restructuring of GBs

3.5 Effect of GB Variability on Dislocation Nucleation

3.6 Summary

**4. Chapter 4 GB Models of thermal and mechanical properties**

4.1 Introduction

4.2 Thermo-Kinetic Models of GB Structure-Strength Transitions

4.3 Temperature and Strain-Rate Sensitive Nucleation Models

4.4 Threshold for Slip Penetration with Different GB Structures

4.5 Summary

**5. Chapter 5 Fundamental models of dislocation dynamics**

5.1 Introduction

5.2 Atomistic Dislocation Core Analysis

5.3 Controlled Multi-Stress Dislocation Dynamics with MD

5.4 Conceptual Framework for Dislocation Velocity Harmonics

5.5 Constitutive Model for Dislocation Mobility with Escaig Stress

5.6 Summary

**6. Chapter 6 Computational framework for polycrystal DD modelling**

6.1 Introduction

6.2 GB Structure Modelling and Mesh-Based Partitioning of DD

6.3 Comprehensive Lattice-Based Framework for GB Interactions

6.4 Benchmarking Applications and Model Validation

6.5 Representing Grain Boundaries as Discrete Dislocation Arrays

6.6 Summary

**7. Chapter 7 Summary and conclusions**

7.1 Summative Overview

7.2 Outlook and Recommendations for Future Work

7.3 Conclusion

**8. References**

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Effects of Grain Boundary Structure in Discrete Simulations of Multiscale Dislocation Dynamics