- Grain
boundary energy and curvature in Monte Carlo and
cellular automata simulations of grain boundary motion
- JK Mason
- Acta Materialia 2015;94:162
- Monte Carlo and cellular automata simulations of grain boundary motion generally suffer from insufficient units of measure. This complicates the comparison of simulations with experiments, the consistent implementation of more than one driving force, and the development of models with predictive capabilities. This paper derives the proportionality constant relating the voxel interaction strength to a boundary energy, derives a formula for the boundary curvature, and uses the Turnbull expression to find the boundary velocity. Providing units of measure for the boundary energy and the boundary curvature allow Monte Carlo simulations and cellular automata simulations, respectively, to be subject to more than one driving force. Using the Turnbull expression to relate a driving pressure to a boundary velocity allows the remaining quantities in cellular automata simulations to be endowed with units of measure. The approach in this paper does not require any calibration of parametric links, but assumes that the voxel interaction strength is a Gaussian function of the distance. The proposed algorithm is implemented in a cellular automata simulation of curvature-driven grain growth.

- Kinetics and
anisotropy of the Monte Carlo model of grain growth
- JK Mason, J Lind, SF Li, BW Reed, M Kumar
- Acta Materialia 2015;82:155
- The Monte Carlo model is one of the most frequently used approaches to simulate grain growth, and retains a number of features that derive from the closely related Ising and Potts models. The suitability of these features for the simulation of grain growth is examined, and several modifications to the Hamiltonian and transition probability function are proposed. The resulting model is shown to not only reproduce the usual behaviors of grain growth simulations, but to substantially reduce the effect of the underlying pixel lattice on the microstructure as compared to contemporary simulations.

- Quadruple
nodes and grain boundary connectivity in three
dimensions
- SF Li, JK Mason, J Lind, M Kumar
- Acta Materialia 2014;64:220
- Recent High-Energy Diffraction Microscopy (HEDM) experiments allow a microstructure to be reconstructed as a 3D volume mesh at a resolution significantly smaller than the characteristic grain size. This is used an as opportunity to evaluate the performance of stereological predictors of the distribution of quadruple node types. The reconstructed microstructures of two materials with different processing histories are found to contain different distributions of quadruple node types, and provide reference points for a comparison of the stereological predictors. While none of the predictors considered here is completely satisfactory, one based on the examination of triangular grains on planar sections and one based on the identification of topological transitions in the grain boundary network on adjacent planar sections perform well enough to be of some practical use. Some of the sources of statistical and systematic error that cause the predictors to deviate from the observed distribution of quadruple node types are explored, and the Hellinger distance is proposed as a means to compare distributions of quadruple node types in practice.

- Statistical
topology of three-dimensional Poisson-Voronoi cells
and cell boundary networks
- EA Lazar, JK Mason, RD MacPherson, DJ Srolovitz
- Physical Review E 2013;88:063309
- Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simulated Poisson-Voronoi structures containing a total of 250000000 cells to provide topological and geometrical statistics of this important class of networks. We also report correlations between some of these topological and geometrical measures. Using these results, we are able to corroborate several conjectures regarding the properties of three-dimensional Poisson-Voronoi networks and refute others. In many cases, we provide accurate fits to these data to aid further analysis. We also demonstrate that topological measures represent powerful tools for describing cellular networks and for distinguishing among different types of networks.

- Convergence
of the hyperspherical harmonic expansion for
crystallographic texture
- JK Mason, OK Johnson
- Journal of Applied Crystallography 2013;46:1722
- Advances in instrumentation allow a material texture to be measured as a collection of spatially-resolved crystallite orientations rather than as a collection of pole figures. The hyperspherical harmonic expansion of a collection of spatially-resolved crystallite orientations is subject to significant truncation error though, resulting in ringing artifacts (spurious oscillations around sharp transitions) and false peaks in the orientation distribution function. This paper finds that the ringing artifacts and the accompanying regions of negative probability density may be mitigated or removed entirely by modifying the coefficients of the hyperspherical harmonic expansion by a simple multiplicative factor. An addition theorem for the hyperspherical harmonics is derived as an intermediate result.

- Statistics
of twin-related domains and the grain boundary network
- JK Mason, OK Johnson, BW Reed, SF Li, JS Stolken, M Kumar
- Acta Materialia 2013;61:6524
- The twin-related domain, or a collection of contiguous grains related by twinning operations, is proposed as the basis for the analysis of grain boundary network connectivity in materials prone to annealing twinning. The distribution of the number of grains in a twin-related domain was measured for materials with a variety of compositions and processing histories. The Weibull distribution is found to accurately reflect many features of the twin-related domain populations, and the parameters of the Weibull distribution vary systematically with the number fraction of resistant boundaries in the microstructure. An alternative model based on the microstructural effects of sequential thermomechanical processing is proposed. This provides an overall fit to the experimental data of comparable quality to the Weibull distribution, while allowing an interpretation of the model parameters that suggests a refinement of the usual thermomechanical processing schedule.

- Topological
view of the thermal stability of nanotwinned copper
- T LaGrange, BW Reed, M Wall, J Mason, T Barbee, M Kumar
- Applied Physics Letters 2013;102:011905
- Sputter deposited nanotwinned copper (nt-Cu) foils typically exhibit strong {111} fiber textures and have grain boundary networks (GBN) consisting of high-angle and a small fraction of low-angle columnar boundaries interspersed with crystallographically special boundaries. Using a transmission electron microscope based orientation mapping system with sub-nanometer resolution, we have statistically analyzed the GBN in as-deposited and annealed nt-Cu foils. From the observed grain boundary characteristics and network evolution during thermal annealing, we infer that triple junctions are ineffective pinning sites and that the microstructure readily coarsens through thermal-activated motion of incoherent twin segments followed by lateral motion of high-angle columnar boundaries.

- Statistical
topology of cellular networks in two and three
dimensions
- JK Mason, EA Lazar, RD MacPherson, DJ Srolovitz
- Physical Review E 2012;86:051128
- Cellular networks may be found in a variety of natural contexts, from soap foams to biological tissues to grain boundaries in a polycrystal, and the characterization of these structures is therefore a subject of interest to a range of disciplines. An approach to describe the topology of a cellular network in two and three dimensions is presented. This allows for the quantification of a variety of features of the cellular network, including a quantification of topological disorder and a robust measure of the statistical similarity or difference of a set of structures. The results of this analysis are presented for numerous simulated systems including the Poisson-Voronoi and the steady-state grain growth structures in two and three dimensions.

- Improved
representation of misorientation information for grain
boundary science and engineering
- S Patala, JK Mason, CA Schuh
- Progress in Materials Science 2012;57:1383
- For every class of polycrystalline materials, the scientific study of grain boundaries as well as the increasingly widespread practice of grain boundary engineering rely heavily on visual representation for the analysis of boundary statistics and their connectivity. Traditional methods of grain boundary representation drastically simplify misorientations into discrete categories such as coincidence vs. non-coincidence boundaries, special vs. general boundaries, and low- vs. high-angle boundaries. Such rudimentary methods are used either because there has historically been no suitable mathematical structure with which to represent the relevant grain boundary information, or, where there are existing methods they are extremely unintuitive and cumbersome to use. This review summarizes recent developments that significantly advance our ability to represent a critical part of the grain boundary space: the misorientation information. Two specific topics are reviewed in detail, each of which has recently enjoyed the development of an intuitive and rigorous framework for grain boundary representation: (i) the mathematical and graphical representation of grain boundary misorientation statistics, and (ii) colorized maps or micrographs of grain boundary misorientation. At the outset, conventions for parameterization of misorientations, projections of misorientation information into lower dimensions, and sectioning schemes for the misorientation space are established. Then, the recently developed hyperspherical harmonic formulation for the description of orientation distributions is extended to represent grain boundary statistics. This allows an intuitive representation of the distribution functions using the axis?angle parameterization that is physically related to the boundary structure. Finally, recently developed coloring schemes for grain boundaries are presented and the color legends for interpreting misorientation information are provided. This allows micrographs or maps of grain boundaries to be presented in a colorized form which, at a glance, reveals all of the misorientation information in an entire grain boundary network, as well as the connectivity among different boundary misorientations. These new and improved methods of representing grain boundary misorientation information are expected to be powerful tools for grain boundary network analysis as the practice of grain boundary engineering becomes a routine component of the materials design paradigm.

- Complete
topology of cells, grains, and bubbles in
three-dimensional microstructures
- EA Lazar, JK Mason, RD MacPherson, DJ Srolovitz
- Phyiscal Review Letters 2012;109:095505
- We introduce a general, efficient method to completely describe the topology of individual grains, bubbles, and cells in three-dimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of three-dimensional microstructures that are often regarded as close analogues in the literature: one resulting from normal grain growth (mean curvature flow) and another resulting from a random Poisson-Voronoi tessellation of space. Grain growth strongly favors particular grain topologies, compared with the Poisson-Voronoi model. Moreover, the frequencies of highly symmetric grains are orders of magnitude higher in the grain growth microstructure than they are in the Poisson-Voronoi one. Grain topology statistics provide a strong, robust differentiator of different cellular microstructures and provide hints to the processes that drive different classes of microstructure evolution.

- A geometric
formulation of the law of Aboav-Weaire in two and
three dimensions
- JK Mason, R Ehrenborg, EA Lazar
- Journal of Physics A: Mathematical and Theoretical 2012;45:065001
- The law of Aboav-Weaire is a simple mathematical expression deriving from empirical observations that the number of sides of a grain is related to the average number of sides of the neighboring grains, and is usually restricted to natural two-dimensional microstructures. Numerous attempts have been made to justify this relationship theoretically, or to derive an analogous relation in three dimensions. This paper provides several exact geometric results with expressions similar to that of the usual law of Aboav-Weaire, though with additional terms that may be used to establish when the law of Abaov-Weaire is a suitable approximation. Specifically, we derive several local relations that apply to individual grain clusters, and a corresponding global relation that is identical in two and three dimensions except for a single parameter [zeta]. The derivation requires the definition and investigation of the average excess curvature, a previously unconsidered physical quantity. An approximation to our exact result is compared to the results of extensive simulations in two and three dimensions, and we provide a compact expression that strikes a balance between complexity and accuracy.

- Computational
topology for configuration spaces of hard disks
- G Carlsson, J Gorham, M Kahle, J Mason
- Physical Review E 2012;85:019905
- We explore the topology of configuration spaces of hard disks experimentally and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik, and Kahle [2] that critical points correspond to configurations of disks with balanced mechanical stresses and suggest conjectures about the asymptotic topology as the number of disks tends to infinity.

- A more
accurate three-dimensional grain growth algorithm
- EA Lazar, JK Mason, RD MacPherson, DJ Srolovitz
- Acta Materialia 2011;59:6837
- In a previous paper, the authors described a simulation method for the evolution of two-dimensional cellular structures by curvature flow that satisfied the von Neumann-Mullins relation with high accuracy. In the current paper, we extend this method to three-dimensional systems. This is a substantial improvement over prior simulations for two reasons. First, this method satisfies the MacPherson-Srolovitz relation with high accuracy, a constraint that has not previously been explicitly implemented. Second, our front-tracking method allows us to investigate topological properties of the systems more naturally than other methods, including Potts models, phase-field methods, cellular automata, and even other front-tracking methods. We demonstrate this method to be feasible in simulating large systems with as many as 100,000 grains, large enough to collect significant statistics well after the systems have reached steady state.

- Expressing
crystallographic textures through the orientation
distribution function: conversion between the
generalized spherical harmonic and hyperspherical
harmonic expansions
- JK Mason, CA Schuh
- Metallurgical and Materials Transactions A 2009;40:2590
- In the analysis of crystallographic texture, the orientation distribution function (ODF) of the grains is generally expressed as a linear combination of the generalized spherical harmonics. Recently, an alternative expansion of the ODF, as a linear combination of the hyperspherical harmonics, has been proposed, with the advantage that this is a function of the angles that directly describe the axis and angle of each grain rotation, rather than of the Euler angles. This article provides the formulas required to convert between the generalized spherical harmonics and the hyperspherical harmonics, and between the coefficients appearing in their respective expansions of the ODF. A short discussion of the phase conventions surrounding these expansions is also presented.

- The
generalized Mackenzie distribution: disorientation
angle distributions for arbitrary textures
- JK Mason, CA Schuh
- Acta Materialia 2009;57:4186
- A general formulation for the disorientation angle distribution function is derived. The derivation employs the hyperspherical harmonic expansion for orientation distributions, and an explicit solution is presented for materials with cubic crystal symmetry and arbitrary textures. The result provides a significant generalization to the well-known Mackenzie distribution function [Mackenzie JK. Biometrika 1958;45:229] for materials with random crystal orientations. This derivation also demonstrates that the relatively new hyperspherical harmonic expansion provides access to results that have been inaccessible with the more traditional "generalized spherical harmonic" expansion that is in current use throughout the field.

- The
Relationship of the Hyperspherical Harmonics to SO(3),
SO(4) and Orientation Distribution Functions
- JK Mason
- Acta Crystallographica A 2009:65:259
- The expansion of an orientation distribution function as a linear combination of the hyperspherical harmonics suggests that the analysis of crystallographic orientation information may be performed entirely in the axis-angle parameterization. Practical implementation of this requires an understanding of the properties of the hyperspherical harmonics. An addition theorem for the hyperspherical harmonics and an explicit formula for the relevant irreducible representatives of SO(4) are provided. The addition theorem is useful for performing convolutions of orientation distribution functions, while the irreducible representatives enable the construction of symmetric hyperspherical harmonics consistent with the crystal and sample symmetries.

- Hyperspherical
harmonics for the representation of crystallographic
texture
- JK Mason, CA Schuh
- Acta Materialia 2008;56:6141
- The feasibility of representing crystallographic textures as quaternion distributions by a series expansion method is demonstrated using hyperspherical harmonics. This approach is refined by exploiting the sample and crystal symmetries to perform the expansion more efficiently. The properties of the quaternion group space encourage a novel presentation of orientation statistics, simpler to interpret than the usual methods of texture representation. The result is a viable alternative to the Euler angle approach to texture standard in the literature today.

- Correlated
grain-boundary distributions in two-dimensional
networks
- JK Mason, CA Schuh
- Acta Crystallographica A 2007;63:315
- In polycrystals, there are spatial correlations in grain-boundary species, even in the absence of correlations in the grain orientations, due to the need for crystallographic consistency among misorientations. Although this consistency requirement substantially influences the connectivity of grain-boundary networks, the nature of the resulting correlations are generally only appreciated in an empirical sense. Here a rigorous treatment of this problem is presented for a model two-dimensional polycrystal with uncorrelated grain orientations or, equivalently, a cross section through a three-dimensional polycrystal in which each grain shares a common crystallographic direction normal to the plane of the network. The distribution of misorientations [theta], boundary inclinations [varphi] and the joint distribution of misorientations about a triple junction are derived for arbitrary crystal symmetry and orientation distribution functions of the grains. From these, general analytical solutions for the fraction of low-angle boundaries and the triple-junction distributions within the same subset of systems are found. The results agree with existing analysis of a few specific cases in the literature but present a significant generalization.

- Determining
the activation energy and volume for the onset of
plasticity during nanoindentation.
- JK Mason, AC Lund, CA Schuh
- Physical Review B 2006;73:054102
- Nanoindentation experiments are performed on single crystals of platinum, and the elastic-plastic transition is studied statistically as a function of temperature and indentation rate. The experimental results are consistent with a thermally activated mechanism of incipient plasticity, where higher time-at-temperature under load promotes yield. Using a statistical thermal activation model with a stress-biasing term, the data are analyzed to extract the activation energy, activation volume, and attempt frequency for the rate-limiting event that controls yield. In addition to a full numerical model without significant limiting assumptions, a simple graphical approximation is also developed for quick and reasonable estimation of the activation parameters. Based on these analyses, the onset of plasticity is believed to be associated with a heterogeneous process of dislocation nucleation, with an atomic-scale, low-energy event as the rate limiter.

- Quantitative
insight into dislocation nucleation from
high-temperature nanoindentation experiments.
- CA Schuh, JK Mason, AC Lund
- Nature Materials 2005;4:617
- Nanoindentation has become ubiquitous for the measurement of mechanical properties at ever-decreasing scales of interest, including some studies that have explored the atomic-level origins of plasticity in perfect crystals. With substantial guidance from atomistic simulations, the onset of plasticity during nanoindentation is now widely believed to be associated with homogeneous dislocation nucleation. However, to date there has been no compelling quantitative experimental support for the atomic-scale mechanisms predicted by atomistic simulations. Our purpose here is to significantly advance the quantitative potential of nanoindentation experiments for the study of dislocation nucleation. This is accomplished through the development and application of high-temperature nanoindentation testing, and the introduction of statistical methods to quantitatively evaluate data. The combined use of these techniques suggests an unexpected picture of incipient plasticity that involves heterogeneous nucleation sites, and which has not been anticipated by atomistic simulations.

- Representations
of Texture
- JK Mason, CA Schuh
- AJ Schwartz, M Kumar, BL Adams, DP Field, editors
- Electron Backscatter Diffraction in Materials Science. Springer, 2009.
- The field of materials science and engineering is fundamentally concerned with manipulating the microstructure of materials in order to control their properties. Recent improvements in instrumentation, which include electron backscatter diffraction (EBSD), dramatically enhance our abilities in this regard by providing extensive crystallographic orientation information of a given two-dimensional section of a microstructure. As this technique has been developed and combined with chemical analysis and serial sectioning methods, it has become possible to access complete three-dimensional chemistry, phase, and crystal orientation information; in short, the microstructural state of a polycrystal may now be completely quantified.
- While these techniques allow an unprecedented opportunity for examining the microstructure-property relationships of materials, in many cases the controlling physics depend on only a relatively small subset of the available information. For example, a variety of the effective tensor properties of materials (e.g. elasticity and transport coefficients) as well as inherently anisotropic, nonlinear properties at the single crystal level (e.g. plasticity and cracking) are governed primarily by the crystallographic texture, or the distribution of crystal orientations within a polycrystal. When properties of this type are of interest, it may be reasonable to simplify the analysis of a material by neglecting the spatial information of the microstructure and examining only the orientations of the crystalline grains.

- Quaternionic
Harmonic Analysis of Texture
- Download
- This is a collection of MATLAB scripts and functions used to manipulate and visualize crystallographic texture information represented as a probability distribution of normalized quaternions on the unit 3-sphere. They are primarily intended as a practial implementation of the mathematics developed in my dissertation. The code is distributed under the GPLv2.

- Front Tracking
Grain Growth 2D
- Download
- This is a C++ program that simulates grain grain in a two-dimensional microstructure, assuming uniform grain boundary energies and mobilities (the simulation is performed in reduced units). The current version has been shown to be able to handle up to ten million grains (memory providing), and is parallelized using OpenMP for shared memory machines. The code is distributed under the GPLv3.