Construction of Polycrystalline Microstructures

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Construction ofPolycrystalline Microstructures

• Dr. Dana Zöllner
• Fakultät für Naturwissenschaften
• Otto-von-Guericke-Universität
• Magdeburg

Workshop in Bad Helmstedt, 24.11.2006
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Introduction
solid materials
crystalline
polycrystalline
amorphous
defects
line defects
point defects
planar defects
e.g. grain boundaries interfaces separating
two crystals of different orientations
grain structure
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Introduction

• control of grain microstructure is key to improve
  materials properties in processing
• prediction of grain structures and their temporal
  evolution due to processes like recrystallization
  and grain growth
• microstructure influences properties in
  materials, like
• crystal plasticity, fracture, corrosion,
  strength or toughness
• different construction methods using

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Using Real Microstructures
existing microstructures with physical
meaning many properties given (e.g. size,
topology, orientation)
- many materials to realistic e.g.
inclusions, two phases - large experimental
expense to get a single structure - limited
number of 2D samples yield uncertainties in
reconstruction of 3D structure!
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ComparisonConstructed Structures vs.
Experimental Results
experimental results - low-carbon steel - by
Liu, Yu and Qin Mat. Sci. Eng. A326 (2002), p276
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Using Mono-dispersive Grain Size Models
? size distribution (lognormal) ? delta
function ? replace grains of different shape and
size by regular hexagon-base columns, cubes or
other equi-sized polyhedra
simple handling fast procedure - discount
any real grain shapes and sizes ? far from reality
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Using Voronoi Tessellations
blue dots initial seed in melt black circles
growing crystals
? crystals connect ? ideal grain ensemble
Poisson Voronoi tessellation
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Using Voronoi Tessellations

• P is set of n distinct points (sites) in space
• Voronoi(P) is subdivision of space in n cells

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Using Voronoi Tessellations

• P is set of n distinct points (sites) in space
• Voronoi(P) is subdivision of space in n cells
• point S is in the cell corresponding to Pi if
  distance(S, Pi) lt distance(S, Pj)

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Using Voronoi Tessellations

• case 1 site
• trivial case
• Idea from http//nms.csail.mit.edu/aklmiu/6.838/
  L7.pdf

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Using Voronoi Tessellations

• case 2 sites
• ? 2 half planes

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Using Voronoi Tessellations

• case 2 sites
• ? 2 half planes

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Using Voronoi Tessellations

• case n collinear sites
• ? n half planes

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Using Voronoi Tessellations

• case 3 non-collinear sites
• ? 3 half edges meeting in a vertex

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Using Voronoi Tessellations

• case 3 non-collinear sites
• ? 3 half edges meeting in a vertex

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Using Voronoi Tessellations

• case 4 non-collinear sites
• ? simple Voronoi diagram

? 3 vertices ? 3 edges ? 3 half edges ? 1
bounded cell ? 3 unbounded cells
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Using Voronoi Tessellations

• ?
• Which of the following statements is true for
• 2D Voronoi diagrams?
• Four non-collinear sites are
• sufficient to create a bounded cell.
• necessary to create a bounded cell.
• 1. and 2.
• none of above.

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Using Voronoi Tessellations
very fast and simple procedure to construct in
2D and 3D many different grain structures
grains look statistically similar to experimental
results
- topological and statistical properties are far
from reality for example number of triple
junctions, triple junction angles, number of
neighbouring grains and grain size distribution
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ComparisonConstructed Structures vs.
Experimental Results
Voronoi tessellation
scaling!
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ComparisonConstructed Structures vs.
Experimental Results
Voronoi tessellation
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Using Size-Distributed-Sphere-Packing

• algorithm
• 1. choice of grain size dis-tribution function f
• 2. calculation of N circles, which follow given
  size distribution f
• 2. arrangement of the cir-cles
• 3. filling of the gaps
• ? 3D works analogously

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Using Size-Distributed-Sphere-Packing
fast and simple procedure to construct in 2D
and 3D many different grain structures grains
look statistically similar to experimental
results size distribution can be chosen, e.g.
from experiment ? nearer to reality than Voronoi
and MDGSM grain boundaries are slightly
curved only three grains meet in a vertex
(usually) triple junction angles near 120
number of neighbouring grains matches better
experimental results
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ComparisonConstructed Structures vs.
Experimental Results
Size-Distributed-Sphere-Packing Rayleigh
Distribution
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ComparisonConstructed Structures vs.
Experimental Results
Size-Distributed-Sphere-Packing Rayleigh
Distribution
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ComparisonConstructed Structures vs.
Experimental Results
Size-Distributed-Sphere-Packing Log-normal
Distribution
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ComparisonConstructed Structures vs.
Experimental Results
Size-Distributed-Sphere-Packing Log-normal
Distribution
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ComparisonConstructed Structures vs.
Experimental Results
Size-Distributed-Sphere-Packing Normal
Distribution
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ComparisonConstructed Structures vs.
Experimental Results
Size-Distributed-Sphere-Packing Normal
Distribution
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Conclusions

• many different methods for constructing 3D grain
  microstructures
• Voronoi tessellation is widely used, but for
  creating polycrystalline structures topological
  and statistical properties are far from reality
• Size-Distributed-Sphere-Packing yields grain
  microstructures matching experimental results

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Thank you for your attention!