Supplementary Material: Mesoscale modeling of polycrystalline light transmission

Lukasz Kuna,1, a) John Mangeri,2 Edward P. Gorzkowski,3 James A. Wollmershauser,3

and Serge Nakhmanson4, 1, b)

1)Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA

2)Institute of Physics, Academy of Sciences of the Czech Republic Na Slovance 2,

182 21 Prague 8, Czech Republic

3)Division of Materials Science and Technology, Naval Research Laboratory,

Washington, DC 20375, USA

4)Department of Materials Science & Engineering and Institute of Material Science,

University of Connecticut, Storrs, Connecticut 06269, USA

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STRUCTURAL MODELS OF POLYCRYSTALLINE MATERIALS

In this study, an open-source computational toolchain is utilized to make predictive mod-

elling of polycrystalline light transmission possible. The toolchain starts with the polycrys-

talline tesselation and mesh generator Neper.1 As shown in Fig. 1, Neper provides the

capability of generating structures that are representative of real polycrystalline ceramic

structures. In this study we employed the centroidal Voronoi as it most closely resembles

a true polycrystalline structure. Fig. 1(b) provides a unique visualization angle of the 3D

mesh, further supporting the similarities between such a mesh and crystal structures that

have been imaged in literature.2–4 While discussing the polycrystalline meshes, it is also

important to consider how the average grain size is determined for a generated mesh. As

shown in Fig. 1(c), the normalized grain size frequency is slightly shifted from unity. The

factor for conversion from a 2D average grain size to 3D is presented as 8/9, and is applied

to the average grain size considerations in this study.5,6

Equally important as morphology, the orientations of the individual grains throughout

the structure must also be representative of real polycrystalline samples. The crystallo-

graphic structure of each material is an important consideration in developing models for

transmission in ceramics and is handled by the combination of open-source software as pre-

sented in this study. To achieve this, the MTEX toolbox7 for modeling crystallographic

textures was used. Utilizing just the crystal and specimen symmetry, MTEX outputs grain

orientations representative of real structures in the form of Bunge Euler angle rotations

(Z1(θ) X2(φ) Z3(ψ)). MTEX is a powerful tool that not only properly normalizes such

rotations,8 but also provides capabilities for analyzing sample orientation distributions. As

displayed in Fig. 2, the materials investigated in this study have slightly different distribu-

tions of misorientation angles which could be a source of disagreement between the models

presented in this work. As one approximation may work well for a certain material, such

as approximating average birefringence to be 2/3∆nmax as is done in the RGD model for

Al2O3, it may be inadequate for application in a different material, as is seen for RGD in

MgF2. Sample pole figures for both Al2O3 (hexagonal) and MgF2 (tetragonal) are displayed

in Fig. 3 and Fig. 4 respectively.

The last major contributors to the toolchain are Ferret9 and MOOSE10. Utilizing the new

optical capabilities within Ferret together with the elastic solver in MOOSE the mean grain-

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FIG. 1. The full block (a) and a structure revealing cut (b) of a 300 grain polycrystalline block

as generated by Neper via a centroidal Voronoi algorithm. The normalized grain distribution in

(c) shows the relationship between average grain size and what the distribution of the grain sizes

is in a polycrystal mesh as generated by Neper.

to-grain birefringence is determined for Al2O3 and MgF2. In the real in-line transmission

(RIT) plots presented in the paper, there are uncertainty lines (orange) included with the

theoretical prediction (green). To keep the calculations relatively inexpensive, but still

accurate, multiple 300 grain models were simulated as opposed to huge thousand grain

models. The uncertainties presented in the RIT plots were determined from the standard

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deviation of 30 different 300 grain configurations as generated by Neper and MTEX.

This was done to eliminate any possible inadvertent texturing in the generation of grain

orientations and to produce a data set large enough to be representative of a polycrystalline

structure.

FIG. 2. Density of misorientation angles for the two different materials, Al2O3 in blue and MgF2 in

red, investigated in this study. There is a higher average frequency of larger misorientation angles

in Al2O3 which suggests that a material with a hexagonal crystal structure will have a higher mean

grain-to-grain birefringence as compared to a tetragonal crystal structure.

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FIG. 3. C-plane (0001), A-plane (112̄0), and R-plane (1̄012) pole figures illustrating the orientation

distribution for three (out of twenty) different samples generated for Al2O3. The pole figures

confirm absence of strong crystallographic texture. Scale is plotted as Multiple of Random Density

(MRD).

REFERENCES

alukasz.kuna@uconn.edu

bserge.nakhmanson@uconn.edu

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FIG. 4. Pole figures for the (100), (110), (111) planes illustrating the orientation distribution for

three (out of twenty) different samples generated for MgF2. The pole figures confirm absence of

strong crystallographic texture. Scale is plotted as Multiple of Random Density (MRD).

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9Ferret is an open-source module for the MOOSE software package and is available at

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10The open-source MOOSE project is developed by Idaho National Laboratory and is avail-

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