A Boltzmann-type kinetic model for misorientation distribution functions in two-dimensional fiber-texture polycrystalline grain growth

Abstract

For mathematical modeling of polycrystalline materials, it is critical to understand how the statistics of evolving grain boundary networks depend on the set of laws that govern the dynamics at a microscopic scale. Such rules describe the motion of grain boundaries and their junctions, as well as criteria for the corresponding topological transitions. Although these laws have been known for quite some time, their precise role in the development of the networks' mesoscopic properties is not sufficiently clear. A possible way to establish a connection between statistical features of the whole networks and fundamental rules for the evolution of individual grain boundaries is to conduct numerical experiments via generally recognized large-scale simulation approaches. Alternatively, it is possible to develop less computationally expensive techniques such as kinetic models based on partial differential equations, while preserving essential characteristics of grain boundary networks. The aim of this paper is to develop a novel kinetic modeling approach for estimating number-and length-weighted misorientation distribution functions in the particular technologically important case of polycrystalline thin films with fiber textures and with anisotropic grain boundary energy densities. Our approach is tested by comparing the numerical results, which it gives for two specific benchmark examples, with the corresponding large-scale simulation results from the related literature. One auxiliary benchmark example is also considered.

A kinetic approach to modeling general-texture evolution in two-dimensional polycrystalline grain growth

“…Now, according to [1,14,18], introduce the number-and length-weighted disorientation distribution functions (DDFs):…”

“…In this situation, the further reduction in the interface energy can be reached only by decreasing the total measure of grain boundaries; see, for instance, [8,9]. Important statistical descriptors of grain boundary networks are misorientation distribution functions (often referred to as grain boundary character distributions), which show relative measures of interfaces with given misorientation parameters; see [1,[9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In order to investigate how the structure and time evolution of misorientation distribution functions depend on the grain boundary energy anisotropy and set of laws governing the dynamics at a microscopic scale, it is possible to conduct numerical experiments via well-known large-scale simulation approaches.…”

“…The paper [1] modified and combined the two kinetic modeling frameworks from [14,22] as applied to the particular case of polycrystalline thin films with so-called fiber textures. In this sufficiently narrow case, grains have nearly identical orientations in the axial direction perpendicular to a considered film, but random radial orientations in the plane of the film; see also [18].…”

“…The current work is aimed at extending the approach developed in [1] to the two-dimensional case of general textures with no restrictions on possible grain orientations, but under the assumption that the grain boundary energy density is a function of a single variable called disorientation. The disorientation of a grain boundary is defined as the smallest rotation angle from all the corresponding symmetrically equivalent misorientations.…”

“…In order to avoid an enormous computational complexity of treating the general-texture case, the kinetic model is constructed from the very beginning in a specific discretized form with respect to the disorientation variable, as against the fibertexture considerations in [1].…”