Jeremy K. Mason

Topological view of the thermal stability of nanotwinned copper

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.

Complete topology of cells, grains, and bubbles in three-dimensional microstructures.

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.

Statistical topology of three-dimensional Poisson-Voronoi cells and cell boundary networks.

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.

Computational topology for configuration spaces of hard disks.

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 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.

The relationship of the hyperspherical harmonics to SO(3), SO(4) and orientation distribution functions.

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.

A geometric formulation of the law of Aboav?Weaire in two and three dimensions

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 ζ. 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.

Correlated grain-boundary distributions in two-dimensional networks

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 phi 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.

Topological similarity of random cell complexes and applications.

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 long-term properties in simulations of a specific model of a dislocation network do not depend on the implementation of dislocation intersections.

Nanolayering around and thermal resistivity of the water-hexagonal boron nitride interface

The water-hexagonal 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 solid-liquid interatomic potential is proposed based on the experimental energy of the interface. The result is markedly different from that given by Lorentz-Berthelot mixing for the Lennard-Jones 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.

A new interlayer potential for hexagonal boron nitride

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 Lennard-Jones 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 non-equilibrium 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 out-of-plane direction.