2015 
 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 curvaturedriven 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.


2014 
 Quadruple nodes and grain boundary connectivity in three dimensions
 SF Li, JK Mason, J Lind, M Kumar
 Acta Materialia 2014;64:220
 Recent HighEnergy 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.


2013 
 Statistical topology of threedimensional PoissonVoronoi 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 PoissonVoronoi 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 threedimensional PoissonVoronoi 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 spatiallyresolved crystallite orientations rather than as a collection of pole figures. The hyperspherical harmonic expansion of a collection of spatiallyresolved 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 twinrelated domains and the grain boundary network
 JK Mason, OK Johnson, BW Reed, SF Li, JS Stolken, M Kumar
 Acta Materialia 2013;61:6524
 The twinrelated 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 twinrelated 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 twinrelated 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 (ntCu) foils typically exhibit strong {111} fiber textures and have grain boundary networks (GBN) consisting of highangle and a small fraction of lowangle columnar boundaries interspersed with crystallographically special boundaries. Using a transmission electron microscope based orientation mapping system with subnanometer resolution, we have statistically analyzed the GBN in asdeposited and annealed ntCu 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 thermalactivated motion of incoherent twin segments followed by lateral motion of highangle columnar boundaries.


2012 
 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 PoissonVoronoi and the steadystate 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. noncoincidence boundaries, special vs. general boundaries, and low vs. highangle 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 threedimensional 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 threedimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of threedimensional 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 PoissonVoronoi tessellation of space. Grain growth strongly favors particular grain topologies, compared with the PoissonVoronoi model. Moreover, the frequencies of highly symmetric grains are orders of magnitude higher in the grain growth microstructure than they are in the PoissonVoronoi 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 AboavWeaire in two and three dimensions
 JK Mason, R Ehrenborg, EA Lazar
 Journal of Physics A: Mathematical and Theoretical 2012;45:065001
 The law of AboavWeaire 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 twodimensional 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 AboavWeaire, though with additional terms that may be used to establish when the law of AbaovWeaire 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.


2011 
 A more accurate threedimensional 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 twodimensional cellular structures by curvature flow that satisfied the von NeumannMullins relation with high accuracy. In the current paper, we extend this method to threedimensional systems. This is a substantial improvement over prior simulations for two reasons. First, this method satisfies the MacPhersonSrolovitz relation with high accuracy, a constraint that has not previously been explicitly implemented. Second, our fronttracking method allows us to investigate topological properties of the systems more naturally than other methods, including Potts models, phasefield methods, cellular automata, and even other fronttracking 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.


2009 
 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 wellknown 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 axisangle 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.


2008 
 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.


2006 
 Correlated grainboundary distributions in twodimensional networks
 JK Mason, CA Schuh
 Acta Crystallographica A 2007;63:315
 In polycrystals, there are spatial correlations in grainboundary 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 grainboundary 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 twodimensional polycrystal with uncorrelated grain orientations or, equivalently, a cross section through a threedimensional 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 lowangle boundaries and the triplejunction 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 elasticplastic 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 timeattemperature under load promotes yield. Using a statistical thermal activation model with a stressbiasing term, the data are analyzed to extract the activation energy, activation volume, and attempt frequency for the ratelimiting 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 atomicscale, lowenergy event as the rate limiter.


2005 
 Quantitative insight into dislocation nucleation from hightemperature nanoindentation experiments.
 CA Schuh, JK Mason, AC Lund
 Nature Materials 2005;4:617
 Nanoindentation has become ubiquitous for the measurement of mechanical properties at everdecreasing scales of interest, including some studies that have explored the atomiclevel 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 atomicscale 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 hightemperature 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.

