2023 
 Simulated surface diffusion in nanoporous gold and its dependence on surface curvature
 CM Winkeljohn, S Shahriar, E Seker, JK Mason
 Computational Materials Science 2023;230:112430
 The morphological evolution of nanoporous gold is generally believed to be governed by surface diffusion. This work specifically explores the dependence of mass transport by surface diffusion on the curvature of a gold surface. The surface diffusivity is estimated by molecular dynamics simulations for a variety of surfaces of constant mean curvature, eliminating any chemical potential gradients and allowing the possible dependence of the surface diffusivity on mean curvature to be isolated. The apparent surface diffusivity is found to have an activation energy of ~0.74 eV with a weak dependence on curvature, but is consistent with the values reported in the literature. The apparent concentration of mobile surface atoms is found to be highly variable, having an Arrhenius dependence on temperature with an activation energy that also has a weak curvature dependence. These activation energies depend on curvature in such a way that the rate of mass transport by surface diffusion is nearly independent of curvature, but with a higher activation energy of ~1.01 eV. The curvature dependencies of the apparent surface diffusivity and concentration of mobile surface atoms is believed to be related to the expected lifetime of a mobile surface atom, and has the practical consequence that a simulation study that does not account for this finite lifetime could underestimate the activation energy for mass transport via surface diffusion by ~0.27 eV.
 Energetic contributions to deformation twinning in magnesium
 E Kapan, S Alkan, CC Aydiner, JK Mason
 Modelling and Simulation in Materials Science and Engineering 2023;31:075002
 Modeling deformation twin nucleation in magnesium has proven to be a challenging task. In particular, the absence of a heterogeneous twin nucleation model which provides accurate energetic descriptions for twinrelated structures indicates a need to more deeply understand twin energetics. To address this problem, molecular dynamics simulations are performed to follow the energetic evolution of {1012} tension twin embryos nucleating from an asymmetricallytilted grain boundary. The line, surface and volumetric terms associated with twin nucleation are identified. A micromechanical model is proposed where the stress field around the twin nucleus is estimated using the Eshelby formalism, and the contributions of the various twinrelated structures to the total energy of the twin are evaluated. The reduction in the grain boundary energy arising from the change in character of the prior grain boundary is found to be able to offset the energy costs of creating the other interfaces. The defect structures bounding the stacking faults that form inside the twin are also found to possibly have significant energetic contributions. These results suggest that both of these effects could be critical considerations when predicting twin nucleation sites in magnesium.
 Geometric conjecture about phase transitions
 OB Eriçok, JK Mason
 Physical Review E 2023;107:064107
 As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach the steady state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and εmixing time of the configuration spaces of harddisk and hardsphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems and numerical evidence suggests that a discontinuity in the εmixing time coincides with the solidfluid phase transition in the thermodynamic limit.
 Designed Y3+ surface segregation increases stability of nanocrystalline zinc aluminate
 LE Sotelo Martin, NM O’Shea, JK Mason, RHR Castro
 The Journal of Physical Chemistry C 2023;127:4239
 The thermal stability of zinc aluminate nanoparticles is critical for their use as catalyst supports. In this study, we experimentally show that doping with 0.5 mol % Y2O3 improves the stability of zinc aluminate nanoparticles. The dopant spontaneously segregates to the nanoparticle surfaces in a phenomenon correlated with excess energy reduction and the hindering of coarsening. Y3+ was selected based on atomistic simulations on a 4 nm zinc aluminate nanoparticle singularly doped with elements of different ionic radii: Sc3+, In3+, Y3+, and Nd3+. The segregation energies were generally proportional to ionic radii, with Y3+ showing the highest potential for surface segregation. Direct measurements of surface thermodynamics confirmed the decreasing trend in surface energy from 0.99 for undoped to 0.85 J/m2 for Ydoped nanoparticles. Diffusion coefficients calculated from coarsening curves for undoped and doped compositions at 850 °C were 4.8 × 10–12 cm2/s and 2.5 × 10–12 cm2/s, respectively, indicating the coarsening inhibition induced by Y3+ results from a combination of a reduced driving force (surface energy) and decreased atomic mobility.
 Dependence of simulated radiation damage on crystal structure and atomic misfit in metals
 JC Stimac, C Serrao, JK Mason
 Journal of Nuclear Materials 2023;585:154633
 This study investigates the evolution of radiation damage in three metals in the low temperature and high radiant flux regime using molecular statics and a Frenkel pair accumulation method to simulate up to 2.0 displacements per atom. The metals considered include Fe, equiatomic CrCoNi, and a fictitious metal with similar bulk properties to the CrCoNi composed of a single atom type referred to as an Aatom. CrCoNi is found to sustain higher concentrations of dislocations than either the Fe or Aatom systems and more stacking faults than the Aatom system. The results suggest that the difference between the concentrations of vacancies and interstitials is substantially smaller for CrCoNi than the Aatom system, perhaps reflecting that the sink capture radius is smaller in CrCoNi due to the roughened potential energy landscape. A model that partitions the major contributions from defects to the stored energy is described, and serves to highlight a general need for higher fidelity approaches to point defect identification.

2022 
 Energy storage under highrate compression of single crystal tantalum
 JC Stimac, N Bertin, JK Mason, VV Bulatov
 Acta Materialia 2022;239:118253
 When a material is plastically deformed the majority of mechanical work is dissipated as heat, and the fraction of plastic work converted into heat is known as the TaylorQuinney coefficient (TQC). Largescale molecular dynamics simulations were performed of high strain rate compression of singlecrystal tantalum, and the resulting integral and differential TQC values are reported up to true strains of 1.0. A phenomenological model is proposed for the energy stored in the material as a function of plastic strain with an asymptotic limit for this energy defined by the deformation conditions. The model reasonably describes the convergence of TQC values to 1.0 with increasing plastic strain, but does not directly address the physical nature of thermomechanical conversion. This is instead developed in a second more detailed model that accurately accounts for energy storage with two distinct contributions, one being the growing dislocation network and the other the point defect debris left behind by moving dislocations. The contribution of the point defect debris is found to lag behind that of the dislocation network but to be substantial for the highrate straining conditions considered here.
 Foundations of a finite nonequilibrium statistical thermodynamics: Extrinsic quantities
 OB Eriçok, JK Mason
 Journal of Physics A: Mathematical and Theoretical 2022;55:295002
 Statistical thermodynamics is valuable as a conceptual structure that shapes our thinking about equilibrium thermodynamic states. A cloud of unresolved questions surrounding the foundations of the theory could lead an impartial observer to conclude that statistical thermodynamics is in a state of crisis though. Indeed, the discussion about the microscopic origins of irreversibility has continued in the scientific community for more than a hundred years. This paper considers these questions while beginning to develop a statistical thermodynamics for finite nonequilibrium systems. Definitions are proposed for all of the extrinsic variables of the fundamental thermodynamic relation that are consistent with existing results in the equilibrium thermodynamic limit. The probability density function on the phase space is interpreted as a subjective uncertainty about the microstate, and the Gibbs entropy formula is modified to allow for entropy creation without introducing additional physics or modifying the phase space dynamics. Resolutions are proposed to the mixing paradox, Gibbs’ paradox, Loschmidt’s paradox, and Maxwell’s demon thought experiment. Finally, the extrinsic variables of the fundamental thermodynamic relation are evaluated as functions of time and space for a diffusing ideal gas, and the initial and final values are shown to coincide with the expected equilibrium values.
 Formation of Complex Spin Textures in Thermally Demagnetized La0.7Sr0.3MnO3 ArtificialSpinIce Structures
 DY Sasaki, RV Chopdekar, ST Retterer, DY Jiang, JK Mason, MS Lee, Y Takamura
 Physical Review Applied 2022;17:064057
 Artificial spin ices (ASIs) have traditionally been designed such that each nanomagnet possesses a singledomain magnetic configuration that is assumed to be minimally perturbed by interisland dipolar interactions. Using xray photoemission electron microscopy to perform magnetic domain imaging, we study thermally demagnetized La0.7Sr0.3MnO3based brickwork ASI arrays and showed that complex spin textures (CSTs) can be stabilized through an appropriate selection of nanoisland width and interisland spacing. While the width dependence can be explained through the dominance of shape anisotropy in isolated nanoislands, the ASIs we investigate demonstrate a complex dependence on both the nanoisland width and interisland spacing. Micromagnetic simulations reveal that interisland dipolar interactions play a role in the formation of CSTs, which are composed of single and doublevortex states. Energy analysis of the simulations provides an understanding of the system energetics that arises from a delicate balance between intraisland effects (i.e., shape anisotropy and exchange energy) and interisland effects (i.e., dipolar interactions between nearestneighbor nanoislands).
 Quotient maps and configuration spaces of hard disks
 OB Eriçok, JK Mason
 Granular Matter 2022;24:1
 Hard disks systems are often considered as prototypes for simple fluids. In a statistical mechanics context, the hard disk configuration space is generally quotiented by the action of various symmetry groups. The changes in the topological and geometric properties of the configuration spaces effected by such quotient maps are studied for small numbers of disks on a square and hexagonal torus. A metric is defined on the configuration space and the various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as complexes. Critical points of the hard disk potential on a configuration space are associated with changes in the topology of the accessible part of the configuration space as a function of disk radius, are conjectured to be related to the configurational entropy of glassy systems, and could reveal the origins of phase transitions in other systems. The number of critical points and their topological and geometric properties are found to depend on the symmetries by which the configuration space is quotiented.

2021 
 Configuration spaces of hard spheres
 OB Eriçok, K Ganesan, JK Mason
 Physical Review E 2021;104:055304
 Hard sphere systems are often used to model simple fluids. The configuration spaces of hard spheres in a threedimensional torus modulo various symmetry groups are comparatively simple and could provide valuable information about the nature of phase transitions. Specifically, the topological changes in the configuration space as a function of packing fraction have been conjectured to be related to the onset of firstorder phase transitions. The critical configurations for 1 to 12 spheres are sampled using a Morsetheoretic approach, and they are available in an online, interactive database. Explicit triangulations are constructed for the configuration spaces of the two sphere system, and their topological and geometric properties are studied. The critical configurations are found to be associated with geometric changes to the configuration space that connect previously distant regions and reduce the configuration space diameter as measured by the commute time and diffusion distances. The number of such critical configurations around the packing fraction of the solidliquid phase transition increases exponentially with the number of spheres, suggesting that the onset of the firstorder phase transition in the thermodynamic limit is associated with a discontinuity in the configuration space diameter.
 Constant of motion for ideal grain growth in three dimensions
 E Eren, JK Mason
 Physical Review B 2021;104:L140103
 Most metallic and ceramic materials are comprised of spacefilling collections of crystalline grains separated by grain boundaries. While this grain structure has been studied for more than a century, there are few rigorous results regarding its global properties available in the literature. We present a rigorous result for threedimensional grain structures that relates the integral of the Gaussian curvature over the grain boundaries to the numbers of grains and quadruple junctions. The result is numerically verified for a grain structure consisting of periodic truncated octahedra.
 Topological transitions during grain growth on a finite element mesh
 E Eren, JK Mason
 Physical Review Materials 2021;5:103802
 The topological transitions that occur to the grain boundary network during grain growth in a material with uniform grain boundary energies are believed to be known. The same is not true for more realistic materials, since more general grain boundary energies in principle allow many more viable grain boundary configurations. A grain growth simulation for such a material therefore requires a procedure to enumerate all possible topological transitions and select the most energetically favorable one. Such a procedure is developed and implemented here for a microstructure represented by a volumetric finite element mesh. As a specific example, all possible transitions for a typical configuration with five grains around a junction point are enumerated, and some exceptional transitions are found to be energetically similar to the conventional ones even for a uniform boundary energy. A general discrete formulation to calculate grain boundary velocities is used to simulate grain growth for an example microstructure. The method is implemented as a C++ library based on SCOREC, an open source massively parallelizable library for finite element simulations with adaptive meshing.
 Classification of atomic environments via the GromovWasserstein distance
 S Kawano, JK Mason
 Computational Materials Science 2021;188:110144
 Interpreting molecular dynamics simulations usually involves automated classification of local atomic environments to identify regions of interest. Existing approaches are generally limited to a small number of reference structures and only include limited information about the local chemical composition. This work proposes to use a variant of the Gromov–Wasserstein (GW) distance to quantify the difference between a local atomic environment and a set of arbitrary reference environments in a way that is sensitive to atomic displacements, missing atoms, and differences in chemical composition. This involves describing a local atomic environment as a finite metric measure space, which has the additional advantages of not requiring the local environment to be centered on an atom and of not making any assumptions about the material class. Numerical examples illustrate the efficacy and versatility of the algorithm.

2020 
 Distribution of topological types in graingrowth microstructures
 EA Lazar, JK Mason, RD MacPherson, and DJ Srolovitz
 Physical Review Letters 2020;125:015501
 An open question in studying normal grain growth concerns the asymptotic state to which microstructures converge. In particular, the distribution of grain topologies is unknown. We introduce a thermodynamiclike theory to explain these distributions in two and threedimensional systems. In particular, a bendinglike energy is associated to each grain topology, and the probability of observing that particular topology depends the order of an associated symmetry group and a thermodynamiclike constant. We explain the physical origins of this approach and provide numerical evidence in support.
 Statistical topology of bond networks with applications to silica
 B Schweinhart, D Rodney, JK Mason
 Physical Review E 2020;101:052312
 Whereas knowledge of a crystalline material’s unit cell is fundamental to understanding the material’s properties and behavior, there are no obvious analogs to unit cells for disordered materials despite the frequent existence of considerable mediumrange order. This article views a material’s structure as a collection of local atomic environments that are sampled from some underlying probability distribution of such environments, with the advantage of offering a unified description of both ordered and disordered materials. Crystalline materials can then be regarded as special cases where the underlying probability distribution is highly concentrated around the traditional unit cell. The H1 barcode is proposed as a descriptor of local atomic environments suitable for disordered bond networks and is applied with three other descriptors to molecular dynamics simulations of silica glasses. Each descriptor reliably distinguishes the structure of glasses produced at different cooling rates, with the H1 barcode and coordination profile providing the best separation. The approach is generally applicable to any system that can be represented as a sparse graph.
 Continuous and optimally complete description of chemical environments using Spherical Bessel descriptors
 E Kocer, JK Mason, H Ertürk
 AIP Advances 2020;10:015021
 Recently, machine learning potentials have been advanced as candidates to combine the highaccuracy of electronic structure methods withthe speed of classical interatomic potentials. A crucial component of a machine learning potential is the description of local atomic environments by some set of descriptors. These should ideally be invariant to the symmetries of the physical system, twicedifferentiable with respectto atomic positions (including when an atom leaves the environment), and complete to allow the atomic environment to be reconstructed upto symmetry. The stronger condition of optimal completeness requires that the condition for completeness be satisfied with the minimumpossible number of descriptors. Evidence is provided that an updated version of the recently proposed Spherical Bessel (SB) descriptors satisfies the first two properties and a necessary condition for optimal completeness. The Smooth Overlap of Atomic Position (SOAP) descriptorsand the Zernike descriptors are natural counterparts of the SB descriptors and are included for comparison. The standard construction of the SOAP descriptors is shown to not satisfy the condition for optimal completeness and, moreover, is found to be an order of magnitude slowerto compute than that of the SB descriptors.

2019 
 Green–Kubo assessments of thermal transport in nanocolloids based on interfacial effects
 T Akiner, E Kocer, JK Mason, H Ertürk
 Materials Today Communications 2019;20:100533
 Thermal transport in a water–Cu nanocolloid system was investigated using equilibrium molecular dynamics. A systematic analysis of the Green–Kubo calculations is presented to clarify the effect of simulation parameters. Several sources of error were identified and quantified for the thermal conductivity estimations, and the effect of the base fluid potential was investigated. Simulations were carried out with a single copper particle for different diameters and water potentials, and thermal enhancements exceeding both theoretical and experimental results were observed in parallel with some other studies in the literature. The anomalous Green–Kubo thermal enhancement results could be explained by the interfacial dynamics and the neglect of calibrating the interaction potential to satisfy the physicallyobserved energy flow at the interface.
 A novel approach to describe chemical environments in highdimensional neural network potentials
 E Kocer, JK Mason, H Ertürk
 The Journal of Chemical Physics 2019;150:154102
 A central concern of molecular dynamics simulations is the potential energy surfaces that govern atomic interactions. These hypersurfaces define the potential energy of the system and have generally been calculated using either predefined analytical formulas (classical) or quantum mechanical simulations (ab initio). The former can accurately reproduce only a selection of material properties, whereas the latter is restricted to short simulation times and small systems. Machine learning potentials have recently emerged as a third approach to model atomic interactions, and are purported to offer the accuracy of ab initio simulations with the speed of classical potentials. However, the performance of machine learning potentials depends crucially on the description of a local atomic environment. A set of invariant, orthogonal, and differentiable descriptors for an atomic environment is proposed, implemented in a neural network potential for solidstate silicon, and tested in molecular dynamics simulations. Neural networks using the proposed descriptors are found to outperform ones using the Behler–Parinello and smooth overlap of atomic position descriptors in the literature.
 Topological constraint theory for network glasses and glassforming liquids: A rigid polytope approach
 S Sen, J Mason
 Frontiers in Materials 2019;6:213
 A variation of the topological constraint theory is proposed where an atomic network is modeled as a collection of rigid polytopes, and which explicitly distinguishes the bond angle constraints as well as rigid bond angles from flexible ones. The proposed theory allows for direct quantitative estimation of the fraction f of zerofrequency or floppy modes of the network. A preliminary model is proposed to connect the theory to the two key experimental observables that characterize glassforming liquids, i.e., the glass transition temperature Tg and fragility m. The predicted values are tested against the literature data available for binary and ternary chalcogenides in the GeAsSe system. The Tg is related to f in this model by the activation entropy associated with the bond scissionrenewal dynamics that is at the heart of transport and relaxation in glassforming liquids. On the other hand, the large and temperaturedependent conformational entropy contribution of the 1polytopes, i.e., the selenium chain elements in these chalcogenide glassforming liquids, plays a key role in controlling the variation of m with f.

2017 
 Thermal characterization assesment of rigid and flexible water models in a nanogap using molecular dynamics
 T Akıner, J Mason, H Ertürk
 Chemical Physics Letters 2017;687:270
 The thermal properties of the TIP3P and TIP5P water models are investigated using equilibrium and nonequilibrium molecular dynamics techniques in the presence of solid surfaces. The performance of the nonequilibrium technique for rigid molecules is found to depend significantly on the distribution of atomic degrees of freedom. An improved approach to distribute atomic degrees of freedom is proposed for which the thermal conductivity of the TIP5P model agrees more closely with equilibrium molecular dynamics and experimental results than the existing state of the art.
 Roundness of grains in cellular microstructures
 FH Lutz, JK Mason, EA Lazar, RD MacPherson
 Physical Review E 2017;96:023001
 Many physical systems are composed of polyhedral cells of varying sizes and shapes. These structures are simple in the sense that no more than three faces meet at an edge and no more than four edges meet at a vertex. This means that individual cells can usually be considered as simple, threedimensional polyhedra. This paper is concerned with determining the distribution of combinatorial types of such polyhedral cells. We introduce the terms fundamental and vertextruncated types and apply these concepts to the grain growth microstructure as a testing ground. For these microstructures, we demonstrate that most grains are of particular fundamental types, whereas the frequency of vertextruncated types decreases exponentially with the number of truncations. This can be explained by the evolutionary process through which grain growth structures are formed and in which energetically unfavorable surfaces are quickly eliminated. Furthermore, we observe that these grain types are “round” in a combinatorial sense: there are no “short” separating cycles that partition the polyhedra into two parts of similar sizes. A particular microstructure derived from the Poisson–Voronoi initial condition is identified as containing an unusually large proportion of round grains. This microstructure has an average of 14.036 faces per grain and is conjectured to be more resistant to topological change than the steadystate grain growth microstructure.
 Nanolayering around and thermal resistivity of the waterhexagonal boron nitride interface
 T Akıner, JK Mason, H Ertürk
 The Journal of Chemical Physics 2017;147:044709
 The waterhexagonal boron nitride interface was investigated by molecular dynamics simulations. Since the properties of the interface change significantly with the interatomic potential, a new method for calibrating the solidliquid interatomic potential is proposed based on the experimental energy of the interface. The result is markedly different from that given by LorentzBerthelot mixing for the LennardJones parameters commonly used in the literature. Specifically, the extent of nanolayering and interfacial thermal resistivity is measured for several interatomic potentials, and the one calibrated by the proposed method gives the least thermal resistivity.
 Improved conditioning of the Floater–Hormann interpolants
 JK Mason
 arXiv:1706.07776
 The Floater–Hormann family of rational interpolants do not have spurious poles or unattainable points, are efficient to calculate, and have arbitrarily high approximation orders. One concern when using them is that the amplification of rounding errors increases with approximation order, and can make balancing the interpolation error and rounding error difficult. This article proposes to modify the Floater–Hormann interpolants by including additional local polynomial interpolants at the ends of the interval. This appears to improve the conditioning of the interpolants and allow higher approximation orders to be used in practice.
 Stability and motion of arbitrary grain boundary junctions
 JK Mason
 Acta Materialia 2017;125:286
 The Herring condition is known as the stability condition for a junction line, but the stability conditions for junction points are not readily available. This paper derives stability conditions for arbitrary junction points and allows for anisotropic surface energies and line energies. A junction is considered to be stable when the force on a small neighborhood around the junction vanishes. When the force does not vanish, the junction is expected to move. Equations of motion are derived for the nodes belonging to polygonal curves or triangulated surfaces, and allow for junction lines and junction points to contribute to the drag on a node. The accuracy of the equations of motion is evaluated by the relative error in the rate of volume change of an adjoining grain. They are found to be secondorder accurate for nodes on a boundary and firstorder accurate for nodes at a junction. A simulation of boundary motion in two dimensions suggests that the equations of motion are of comparable accuracy to alternatives in the literature, and have the advantage of less computational complexity.


2016 
 A new interlayer potential for hexagonal boron nitride
 T Akıner, JK Mason, H Ertürk
 Journal of Physics: Condensed Matter 2016;28:385401
 A new interlayer potential is developed for interlayer interactions of hexagonal boron nitride sheets, and its performance is compared with other potentials in the literature using molecular dynamics simulations. The proposed potential contains Coulombic and LennardJones 6–12 terms, and is calibrated with recent experimental data including the hexagonal boron nitride interlayer distance and elastic constants. The potentials are evaluated by comparing the experimental and simulated values of interlayer distance, density, elastic constants, and thermal conductivity using nonequilibrium molecular dynamics. The proposed potential is found to be in reasonable agreement with experiments, and improves on earlier potentials in several respects. Simulated thermal conductivity values as a function of the number of layers and of temperature suggest that the proposed LJ 6–12 potential has the ability to predict some phonon behaviour during heat transport in the outofplane direction.
 Topological similarity of random cell complexes and applications
 B Schweinhart, JK Mason, RD MacPherson
 Physical Review E 2015;92:063308
 Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. The various proposals in the literature are usually motivated by the analysis of particular physical systems and do not necessarily apply to general situations. The central concepts in this paper—the swatch and the cloth—provide a description of the local topology of a cell complex that is general (any physical system that can be represented as a cell complex is admissible) and complete (any statistical question about the local topology can be answered from the cloth). Furthermore, this approach allows a distance to be defined that measures the similarity of the local topology of two cell complexes. The distance is used to identify a steady state of a model grain boundary network, quantify the approach to this steady state, and show that the steady state is independent of the initial conditions. The same distance is then employed to show that the longterm properties in simulations of a specific model of a dislocation network do not depend on the implementation of dislocation intersections.


2015 
 Geometric and topological properties of the canonical graingrowth microstructure
 JK Mason, EA Lazar, RD MacPherson, DJ Srolovitz
 Physical Review E 2015;92:063308
 Many physical systems can be modeled as large sets of domains “glued” together along boundaries—biological cells meet along cell membranes, soap bubbles meet along thin films, countries meet along geopolitical boundaries, and metallic crystals meet along grain interfaces. Each class of microstructures results from a complex interplay of initial conditions and particular evolutionary dynamics. The statistical steadystate microstructure resulting from isotropic grain growth of a polycrystalline material is canonical in that it is the simplest example of a cellular microstructure resulting from a gradient flow of an energy that is directly proportional to the total length or area of all cell boundaries. As many properties of polycrystalline materials depend on their underlying microstructure, a more complete understanding of the grain growth steady state can provide insight into the physics of a broad range of everyday materials. In this paper we report geometric and topological features of these canonical two and threedimensional steadystate microstructures obtained through extensive simulations of isotropic grain growth.
 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;46:220
 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;46:1772
 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.

