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Sethuraman Sankaran [email protected]

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Computing grain boundary properties using ab-initio simulations

Sethuraman Sankaran

MAE 715 Course Project

An overview

- Grain boundaries: an atomistic viewpoint
- Grain boundaries : Literature survey
- Modeling grain boundaries : techniques and issues
- Computing properties of grain boundaries
- Doping of twist grain boundaries
- Numerical examples
- Summary and extensions

Grain boundaries: an atomistic viewpoint

Grain boundaries

What is a grain boundary?

A grain boundary occurs as a result of change in the orientations of atoms on either side of the grain boundary

Twist grain boundary (view about normal to the grain boundary)

Tilt grain boundary(view about the plane of the grain boundary)

Since the analysis of twist and tilt grain boundaries is computationally very different, I concentrate on twist grain boundary for this presentation

What happens at a grain boundary?

Arrangement of atoms close to the grain boundary shape themselves to accommodate the reorientation of atoms. Certain atoms along the original structure have to be removed while new atoms have to be added at other sites to obtain the new configuration.

Mathematical representation of grain boundaries

The boundary is defined by 5 parameters: The three rotation angles needed to "produce" grain II, and two parameters to define the boundary plane in the coordinate system of the reference grain I.

- How do we write down a mathematical description of a 5-parameter grain boundary? Answer: there are 3 useful methods.
- Disorientation + plane
- Plane of 1st grain + plane of 2nd grain + twist angle
- Matrix (4 x 4)

CSL: angle from trigonometric considerations: 26.565o

Coinciding site lattice:

A pattern of coincidences is observed for some ‘special’ angles of the grain boundary. Grain boundaries at these special angles are named ‘special grain boundaries’. These have lower energies and considerably higher grain boundary mobilities.

Grain boundaries: Literature review

On the structure of tilt grain boundaries in cubic metals: pure tilt boundaries (Sutton et. al., Phil. Trans. Roy. Soc. 1983)

- One of the earliest works on computer modeling of grain boundaries
- How to model grain boundaries in metals (considered copper and aluminium)
- Few atoms with pair potentials

Miura et.al. 1990

Measured value of grain boundary energies. This shows that the structurally stable configurations are those involving CSL’s. Hence, the grain boundary analysis for CSL angles is an interesting subclass of problems to study.

Grain boundaries of more complex structures

Dawson I and Bristowe PD. First principles study of a tilt grain boundary in rutile, Physical Review B 1996

Different bond lengths and angles for the rutile TiO2 structure makes it a complicated structure to study.

Different bond lengths and angles for the rutile TiO2 structure makes it a complicated structure to study.

Cleri F. Atomic and electronic structure of high-energy grain boundaries in silicon and carbon, CMS 2001.

Grain boundary impurities

- Ogata et. al. Ab-initio analysis of aluminium Sigma=5 grain boundaries – fundamental structures and effects of silicon impurity, CMS, 1997.
- DFT computations of stable structures in Silicon doped Aluminium grain boundaries (a total of 38 atoms was considered).

- Braithwaite JS et al. Grain boundary impurities in iron, Acta Materialia 2005
- Since the size of iron atom is larger than the size of doping ions, it was seen that the gap between grains reduce and the added ions fill some gap in the boundary of the grains

C doped

P doped

Computing grain boundary properties

Comprehensive review paper:

Farkas D. Atomistic theory and computer simulation of grain boundary structure and diffusion, J Phys. Condens. Matter 2000; 12:R497-R516.

- Properties of grain boundaries:
- Grain boundary diffusivity (MD): Molecular dynamics simulations (Liu and coworkers) ‘Molecular statics and molecular dynamics study of diffusion along [001] tilt grain boundaries in Ag’ Physical Review B 1995.
- Self diffusion in copper (MD): Comparative study of grain-boundary migration and grain-boundary self-diffusion of [0 0 1] twist-grain boundaries in copper by atomistic simulations, Acta Materialia 2005.
- Stress strain curves for alumina (ab-initio): Chen J et al. Ab initio theoretical tensile test on Y-doped S=3 grain boundary in Al2O3, Acta Materialia 2005.
- GB diffusivity coefficient (MC) : Sorensen et al. Diffusion mechanisms in Cu grain boundaries, Physical Review B. 2000.

Modeling grain boundaries: techniques and issues

- Analytically compute the grain boundary angle based on the CSL index.
- Develop an uniform grain and choose the layer which forms the grain boundary.
- Operate one layer of grains with the rotation matrix computed from the grain boundary angle.
- Remove or fill atoms until a fit is produced – most crucial step

CSL grain boundaries

Model 2: Sigma=5

Model 1: Sigma=25

Model 2: # atoms = 160

Model 1: # atoms = 800

Modeling issues

- Developing a fit between the atoms is quite difficult. Because of this, there is a certain minimum number of atoms that needs to be modeled
- The size of the supercell cannot be unreasonably small since this implies a frequent repetition of grain boundaries.
- The maximum number of atoms to be modeled is limited by computational time required. For instance, ab-initio computations take around 8 hours to converge to the optimal structure. This necessitates the use of large number of computational clusters for structural optimization (e.g. full ab-initio computation for rutile structure with 360 atoms took one day on a 512 node CRAY cluster).

Properties of grain boundaries

Grain boundary energy

Grain boundary energy is defined as the difference in energies between a supercell with a grain boundary and a perfect lattice containing the same number of atoms divided by the area of the grain boundary.

Volume expansion factor is defined as the volume difference of the bicrystal with that of a perfect crystal containing the same number of atoms divided by the interfacial area.

Grain boundary diffusivity

- GB self-diffusion coefficient

- Self diffusion coefficient is computed by performing a molecular dynamics simulation considering the atoms in different spatial locations according to the orientation of grains
- EAM or Lennard Jones type potentials are usually employed for such simulations

Tensile tests on nanocrystals

Chen et. al.

- Compute the atomic arrangement for a specific grain boundary
- Apply forces along a specific direction in which the properties are to be computed
- When grain boundaries are present, the presence of additional dopant atoms may improve the properties of the specimen
- VASP code with 220 atoms were used for the problem

Modeling doping in grain boundaries

Substitutional doping

Interstitial doping

Dopant atoms in interstices

Substitutional doping: Dopant atoms substitute the parent atoms. Seen when the size of dopants are similar to the size of parent atoms

Interstitial doping: Dopant atoms occupy interstices of the original grain. Occurs when dopant atoms are significantly smaller in size.

Numerical examples

Problem statement

- Predict the optimal structure of Copper twist grain boundaries and their properties by using energy minimization techniques. Various Sigma- twist grain boundary of Copper are analyzed. Energies of the interface are computed as well as the volume expansions and compared with other results

Problem parameters

- Number of atoms: varies from 80 (SIGMA-5) to around 408.
- Interatomic potentials: EAM (Gulp software was utilized).
- Lattice : FCC with a cell size of 3.615 Ao
- Supercell size: varies with the number of atoms

Seki: Used EAM potentials and between 4000-6000 atoms in his simulation of Cu twist boundaries. Comparison shows a good match between results in the Literature and that computed here

Grain boundary volume expansion – Test case

Seki: Comparison of volume expansion due to the presence of grain boundaries.

The figure shows comparison with modified embedded atom potentials. MEAMs have angular dependent terms which takes into account the misorientation between grains. This shows that angular dependency has to be accounted for in such materials.

Problem statement

- Predict the optimal structure of Aluminium twist grain boundaries and their properties by using energy minimization techniques. Predict the effect of dopant Silicon atoms at the grain boundaries of Aluminium

Problem parameters

- Number of atoms: 44 (SIGMA-5) in the presence of doping.
- Interatomic potentials: EAM (Gulp software was utilized).
- Lattice : FCC with a cell size of 3.615 Ao
- Supercell size: varies with the number of atoms

Comparisons: Structure

Figure shows optimal structure that was obtained for Aluminium twist grain boundary viewed across the [001] plane. A good structural match is observed between the calculations.

SIGMA-5 grain boundaries

Initial and final structures of the twist grain boundaries.

Ogata et. al. ab-initio computations

Initial and final configurations are shown in the figure

Structures for other SIGMA boundaries

SIGMA-13 grain boundary

SIGMA-17 grain boundary

Silicon dopants on Aluminium

Coloring scheme

Blue: Silicon Doping on

Red: Parent aluminium atoms

Grain boundary surface

Ogata et. al. (1997) ‘For the computation of substitutional doping of Silicon on Aluminium, an insignificant change of the structure is only noted when the doped atoms are on the grain boundary. Hence results are not plotted.’

Conclusions and extensions

- Structures of twist boundaries were studied for commonly occurring metals such as Copper and Aluminium.
- Considerations of energetics and structures using EAM shows good match with literature.
- A complete ab-initio analysis will require large computing clusters dedicated to structural optimization of such configurations.
- One improvement that can be made is the use of modified embedded atom potentials. This utilized the orientation of corresponding atoms and may involve information that is not available in the embedded atom potentials.
- Diffusivity properties of grain boundaries can be computed by developing energies for a large number of grain boundary structures, using these for interatomic force computations and utilizing a molecular dynamics code that can model diffusion across grain boundaries.

- Parr and coworkers have devised a new scheme for DFT computations

Entropy deficiency constraints:

Optimize entropy:

After some computations, they showed that the DFT solution can be obtained by the minimization of a free energy like functional at large temperatures:

The future..?

Some preliminary results from Parr’s paper conclude (verbatim):

In practical calculations resultant equations from the entropy-deficiency constraint appears to be more convenient in early stages of iteration. More appealingly, the employment of entropy deficiencies produce a converged

density much closer to the exact input density . Further, the new scheme seems to be particularly satisfying from a physical viewpoint. It amounts to the minimizing

with respect to r, at constant u, of a defined Helmholtz-like free-energy functional

Current research in this ares

- Parr and coworkers on development of DFT from entropy constraints
- Nalewajski and coworkers: Development of information theory basis for wave functions (the Hirshfeld partitioning)
- Important point to note: this DFT formalism has not yet been implemented

Thank You