## Microstructure Statistics Property Relations of
Anisotropic

Polydisperse Particulate Composites using Tomography

A. Gillman, K. Matous and S. Atkinson

Department of Aerospace and Mechanical Engineering

University of Notre Dame

Notre Dame, IN, 46556, USA.

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#### Abstract

In this paper, a systematic method is
presented for developing
microstructure-statistics-property relations of
anisotropic polydisperse particulate composites using
micro-computer tomography (micro-CT). Micro-CT is used to
obtain a detailed three-dimensional representation of
polydisperse microstructures, and a novel image processing
pipeline is developed for identifying particles. In this
work, particles are modeled as idealized shapes in order
to guide the image processing steps and to provide a
description of the discrete micro-CT dataset in continuous
Euclidean space. n-point probability functions used to
describe the morphology of mixtures are calculated
directly from real microstructures. The statistical
descriptors are employed in the Hashin-Shtrikman
variational principle to compute overall anisotropic
bounds and self-consistent estimates of the thermal
conductivity tensor. We make no assumptions of statistical
isotropy nor ellipsoidal symmetry, and the statistical
description is obtained directly from micro-CT data.
Various mixtures consisting of polydisperse ellipsoidal
and spherical particles are prepared and studied to show
how the morphology impacts the overall anisotropic thermal
conductivity tensor.

#### Conclusions

In this work, we present a systematic
microstructure characterization procedure anchored in
micro-CT data that is used to establish
microstructure-statistics-property relations of
polydisperse particulate mixtures. A novel image
processing pipeline is developed that accurately
identifies particles while maintaining low errors.
Improvements in the image processing pipeline are achieved
when compared to a traditional technique. For all
compositions considered, the volume losses due to image
segmentation are less than 4%. These low errors indicate
that scientifically sound and repeatable results have been
achieved. Next, we developed a description of the
polydisperse system in continuous Euclidean space. This
idealized representation provides a substantial reduction
in the dataset size and enables easier data manipulation
and understanding.

After characterizing the
microstructure, three-dimensional n-point probability
functions of real polydisperse mixtures are calculated. We
show that second-order probability functions do not
exhibit ellipsoidal nor any other material symmetry.
Therefore, assessment of overall material constants in a
closed form is unattainable.

The statistical description is then
used to compute bounds and self-consistent estimates of
the anisotropic thermal conductivity tensor using the
Hashin-Shtrikman variational principle. This is the first
time to the best of our knowledge, that the anisotropic
second-order estimates of polydisperse composites are
calculated without assumptions on an inclusion’s shape,
configuration and/or material anisotropy. The overall
properties show increasing anisotropy in the overall
thermal conductivity tensor for packs with more transverse
isotropic ellipsoidal inclusions. Moreover, the upper and
lower bounds provide limits on the anisotropy of the
mixtures. Due to the larger amounts of statistical
anisotropy for the semi-ordered mixture, the measure of
anisotropy for the overall conductivity tensor of this
pack was significantly larger than for the randomized one.

#### Acknowledgments

The authors would like to acknowledge
support from IllinoisRocstar LLC under the contract number
FA9300-10-C-3003 (Edwards Air Force Base, SBIR Phase II
project) by the Office of the Secretary of Defense as a
part of the Phase II SBIR program.

Any opinions, findings and conclusions
or recommendations expressed in this material are those of
the author(s) and do not necessarily reflect the views of
IllinoisRocstar LLC and the U.S. Air Force.