Symposia on Quantum Mechanical Models of Materials Sethuraman Sankaran [email protected] MAE 715...

Symposia on Quantum Symposia on Quantum

Mechanical Models of MaterialsMechanical Models of Materials

CCOORRNNEELLLL U N I V E R S I T Y


Sethuraman Sankaran

[email protected]

MAE 715 Course Project

Computing grain boundary properties using ab-initio simulations





An overviewAn 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 boundariesGrain 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





Grain boundary structureGrain boundary structure

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.





Coinciding site latticesCoinciding site lattices

• How do we write down a mathematical description of a 5-parameter grain boundary? Answer: there are 3 useful methods.

1. Disorientation + plane2. Plane of 1st grain + plane of 2nd grain +

twist angle3. Matrix (4 x 4)

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.

CSL: angle from trigonometric considerations: 26.565o





Grain boundaries: Literature review





Literature study : Grain boundary modelingLiterature study : Grain boundary modeling

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 structuresGrain 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.





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).

Grain boundary impuritiesGrain boundary impurities

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 dopedP doped





Computing grain boundary propertiesComputing 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





Steps involved to develop a basic GB modelSteps involved to develop a basic GB model

• Analytically compute the grain boundary angle based on the CSL index.Analytically compute the grain boundary angle based on the CSL index.

• Develop an uniform grain and choose the layer which forms the grain Develop an uniform grain and choose the layer which forms the grain boundary.boundary.

• Operate one layer of grains with the rotation matrix computed from the grain Operate one layer of grains with the rotation matrix computed from the grain boundary angle.boundary angle.

• Remove or fill atoms until a fit is produced – most crucial stepRemove or fill atoms until a fit is produced – most crucial step

CSL grain boundaries





Twist grain boundaries in copperTwist grain boundaries in copper

Model 1: Sigma=25 Model 2: Sigma=5

Model 1: # atoms = 800 Model 2: # atoms = 160





Modeling issuesModeling issues

• Developing a Developing a fit between the atomsfit between the atoms is quite difficult. Because of this, there is is quite difficult. Because of this, there is

a certain minimum number of atoms that needs to be modeleda certain minimum number of atoms that needs to be modeled

• The size of the The size of the supercell cannot be unreasonably smallsupercell cannot be unreasonably small since this implies a since this implies a

frequent repetition of grain boundaries.frequent repetition of grain boundaries.

• The maximum number of atoms to be modeled is limited by computational The maximum number of atoms to be modeled is limited by computational

time required. For instance, ab-initio computations take around 8 hours to time required. For instance, ab-initio computations take around 8 hours to

converge to the optimal structure. This necessitates the use of large number converge to the optimal structure. This necessitates the use of large number

of computational clusters for structural optimization (e.g. of computational clusters for structural optimization (e.g. full ab-initio full ab-initio

computation for rutile structure with 360 atoms took one day on a 512 node computation for rutile structure with 360 atoms took one day on a 512 node

CRAY clusterCRAY cluster).).





Properties of grain boundaries





Grain boundary energyGrain boundary energy

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

GB perfect

GB

E EGBE

A

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

GB perfect

GB

V VVE

A





• GB self-diffusion coefficientGB self-diffusion coefficient

Grain boundary diffusivityGrain boundary diffusivity

Self diffusion coefficient is computed by performing a molecular dynamics Self diffusion coefficient is computed by performing a molecular dynamics simulation considering the atoms in different spatial locations according to the simulation considering the atoms in different spatial locations according to the orientation of grainsorientation of grains

EAM or Lennard Jones type potentials are usually employed for such EAM or Lennard Jones type potentials are usually employed for such simulationssimulations





Tensile tests on nanocrystalsTensile 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 boundariesModeling 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





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

Problem statementProblem statement

Problem parameters

1.1. Number of atoms: varies from 80 (SIGMA-5) to around 408.Number of atoms: varies from 80 (SIGMA-5) to around 408.

2.2. Interatomic potentials: EAM (Gulp software was utilized).Interatomic potentials: EAM (Gulp software was utilized).

3.3. Lattice : FCC with a cell size of 3.615 ALattice : FCC with a cell size of 3.615 Aoo

4.4. Supercell size: varies with the number of atomsSupercell size: varies with the number of atoms





Grain boundary energies – Test caseGrain boundary energies – Test case

Comparison of grain boundary energies

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50

Angle (degrees)

En

erg

y (m

J/m

^2)

Seki's model

Current model

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





Comparison of grain boundary energies

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50

Angle (degrees)

En

erg

y (m

J/m

^2)

Seki's model

Current model

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

Grain boundary volume expansion – Test caseGrain boundary volume expansion – Test case





Modified EAM potentialsModified EAM potentials

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50

Grain boundary angle (degrees)

En

erg

y (m

J/m

^2)

EAM

Modified EAM

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





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

Problem statementProblem statement

Problem parameters

1.1. Number of atoms: 44 (SIGMA-5) in the presence of doping.Number of atoms: 44 (SIGMA-5) in the presence of doping.

2.2. Interatomic potentials: EAM (Gulp software was utilized).Interatomic potentials: EAM (Gulp software was utilized).

3.3. Lattice : FCC with a cell size of 3.615 ALattice : FCC with a cell size of 3.615 Aoo

4.4. Supercell size: varies with the number of atomsSupercell size: varies with the number of atoms





Comparisons: StructureComparisons: Structure

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

Initial and final configurations are shown in the figureInitial and final configurations are shown in the figure

Ogata et. al. ab-initio Ogata et. al. ab-initio computationscomputations

Initial and final structures of Initial and final structures of the twist grain boundaries.the twist grain boundaries.

SIGMA-5 grain boundariesSIGMA-5 grain boundaries





SIGMA-17 grain boundarySIGMA-17 grain boundary SIGMA-13 grain boundarySIGMA-13 grain boundary

Structures for other SIGMA boundariesStructures for other SIGMA boundaries





Silicon dopants on AluminiumSilicon dopants on Aluminium

Grain boundary surface Coloring schemeBlue: Silicon Doping onRed: Parent aluminium atomsOgata 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





ConclusionsConclusions

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.





ExtensionsExtensions

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





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Symposia on Quantum Mechanical Models of Materials Sethuraman Sankaran [email protected] MAE 715...

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