## Bias in polycrystal topology caused by grain boundary motion by mean curvature

##### Authors

Keller, Trevor

##### ORCID

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##### Other Contributors

Lewis, Daniel J.

Cutler, Barbara M.

Huang, Liping

Keblinski, Pawel

Maniatty, Antoinette M.

Cutler, Barbara M.

Huang, Liping

Keblinski, Pawel

Maniatty, Antoinette M.

##### Issue Date

2015-05

##### Keywords

Materials engineering

##### Degree

PhD

##### Terms of Use

Attribution-NonCommercial-NoDerivs 3.0 United States

This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.

This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.

##### Full Citation

##### Abstract

During heat treatment of polycrystalline materials, grain shape affects the rate of grain growth. In 2-D, the von Neumann-Mullins Law requires grains with fewer than six edges to shrink, greater than six edges to grow, and hexagonal grains to be stable regardless of edge lengths or curvatures. The 3-D analogue, described by the MacPherson-Srolovitz relation, does not explicitly depend on any topological feature (number of faces, edges, or vertices), yet there is bias in the observed grain topologies in 3-D metal polycrystals. In order to investigate this bias and determine its origins, numerical simulations of ideal polycrystalline materials, characterization of topologies, and comparisons to possible polyhedral shapes were performed.

By changing the physics of grain growth to decrease edge and face curvature, the population of three-edged faces should increase, with the standard deviation in edges per face increasing proportionally. To test this hypothesis, a phase-field model of grain growth was implemented with lower mobility on triple junctions than on other features. This approach, known as a "vertex drag" model in 2-D, tends to straighten grain edges. From large-scale 2-D simulations, vertex mobility 100x lower than the edge mobility was found to increase the relative proportion of 3-edged grains by 25%. While the effect is small in magnitude, this result supports motion by mean curvature as the root cause of bias in polycrystalline grain topology.

One dataset, a synthetic microstructure with flat edges and faces, was biased more weakly than the rest. The remaining four datasets involved motion by mean curvature, the fundamental mechanism of grain growth, under which interfaces move toward their center of curvature with velocity proportional to that curvature: sharply curved faces move faster than more gently curved ones, and flat faces move not at all. To satisfy force balance at the vertices, three-edged faces in polycrystals become highly curved and quickly collapse during grain growth, but the laws of topology require that grains with between ten and sixteen faces have five edges per face, on average. This span covers the median and mean number of faces in polycrystalline grain populations. Therefore, as three-edged faces collapse, faces with more than 5 edges must also lose edges to maintain grain boundary network connectivity.

Normal grain growth in polycrystalline materials is characterized by a self-similar distribution of topological properties: the average grain area increases with heat treatment time, but the average number of faces per grain remains constant. Therefore, distributions of the number of faces per grain are commonly reported characteristics of polycrystals. To investigate bias in grain topologies, the number of edges per face on each grain in the polycrystal was extracted, then the standard deviation of this quantity was computed for each grain. For grains resembling Platonic solids with equal numbers of edges on each face, such as the Platonic tetrahedron, hexahedron, and dodecahedron, this quantity is zero. In typical grains with more diverse faces, the standard deviation increases. The average, upper, and lower bound of standard deviations possible for all polyhedra with a given number of faces were determined by enumerating each using a graph theory-based code, plantri. Several polycrystalline datasets were then obtained and analyzed: two synthetic, two simulated grain growth, and one experimental reconstruction of titanium. The polycrystals all exhibited lower averages of the standard deviation of edges per face than the enumerated polyhedra, demonstrating bias. Specifically, the bias in grain growth favors more "regular" topologies, with a smaller spread in the number of edges per face than would occur at random.

By changing the physics of grain growth to decrease edge and face curvature, the population of three-edged faces should increase, with the standard deviation in edges per face increasing proportionally. To test this hypothesis, a phase-field model of grain growth was implemented with lower mobility on triple junctions than on other features. This approach, known as a "vertex drag" model in 2-D, tends to straighten grain edges. From large-scale 2-D simulations, vertex mobility 100x lower than the edge mobility was found to increase the relative proportion of 3-edged grains by 25%. While the effect is small in magnitude, this result supports motion by mean curvature as the root cause of bias in polycrystalline grain topology.

One dataset, a synthetic microstructure with flat edges and faces, was biased more weakly than the rest. The remaining four datasets involved motion by mean curvature, the fundamental mechanism of grain growth, under which interfaces move toward their center of curvature with velocity proportional to that curvature: sharply curved faces move faster than more gently curved ones, and flat faces move not at all. To satisfy force balance at the vertices, three-edged faces in polycrystals become highly curved and quickly collapse during grain growth, but the laws of topology require that grains with between ten and sixteen faces have five edges per face, on average. This span covers the median and mean number of faces in polycrystalline grain populations. Therefore, as three-edged faces collapse, faces with more than 5 edges must also lose edges to maintain grain boundary network connectivity.

Normal grain growth in polycrystalline materials is characterized by a self-similar distribution of topological properties: the average grain area increases with heat treatment time, but the average number of faces per grain remains constant. Therefore, distributions of the number of faces per grain are commonly reported characteristics of polycrystals. To investigate bias in grain topologies, the number of edges per face on each grain in the polycrystal was extracted, then the standard deviation of this quantity was computed for each grain. For grains resembling Platonic solids with equal numbers of edges on each face, such as the Platonic tetrahedron, hexahedron, and dodecahedron, this quantity is zero. In typical grains with more diverse faces, the standard deviation increases. The average, upper, and lower bound of standard deviations possible for all polyhedra with a given number of faces were determined by enumerating each using a graph theory-based code, plantri. Several polycrystalline datasets were then obtained and analyzed: two synthetic, two simulated grain growth, and one experimental reconstruction of titanium. The polycrystals all exhibited lower averages of the standard deviation of edges per face than the enumerated polyhedra, demonstrating bias. Specifically, the bias in grain growth favors more "regular" topologies, with a smaller spread in the number of edges per face than would occur at random.

##### Description

May 2015

School of Engineering

School of Engineering

##### Department

Dept. of Materials Science and Engineering

##### Publisher

Rensselaer Polytechnic Institute, Troy, NY

##### Relationships

Rensselaer Theses and Dissertations Online Collection

##### Access

CC BY-NC-ND. Users may download and share copies with attribution in accordance with a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License. No commercial use or derivatives are permitted without the explicit approval of the author.