Abnormal grain growth mediated by fractal boundary migration at the nanoscale

Abstract

Modern engineered materials are composed of space-filling grains or domains separated by a network of interfaces or boundaries. Such polycrystalline microstructures have the capacity to coarsen through boundary migration. Grain growth theories account for the topology of grains and the connectivity of the boundary network in terms of the familiar Euclidian dimension and Euler's polyhedral formula, both of which are based on integer numbers. However, we recently discovered an unusual growth mode in a nanocrystalline Pd-Au alloy, in which grains develop complex, highly convoluted surface morphologies that are best described by a fractional dimension of ∼1.2 (extracted from the perimeters of grain cross sections). This fractal value is characteristic of a variety of domain growth scenarios-including explosive percolation, watersheds of random landscapes, and the migration of domain walls in a random field of pinning centers-which suggests that fractal grain boundary migration could be a manifestation of the same universal behavior.

Figure 1

Microstructural maps of abnormal grain growth in nanocrystalline Pd-10 at% Au, recorded by electron backscatter diffraction (EBSD) following heat treatments of (a) 155 °C for 202 h and (b) 400 °C for 4 h. The speckled contrast visible between single-color micrometer-sized regions in (a) is caused by the presence of grains smaller than the point resolution of the EBSD technique at this magnification (0.15 μm). The perimeters of abnormal grains in (a) and (b) are much rougher and more convoluted than the smooth boundaries seen in (c), which represents the microstructure of a Pd-10 at% Au sample prepared by arc melting and subsequent solidification.

Figure 2

Evaluation of the box-counting fractal dimension D₀, extracted from log-log plots of the number of boxes N(ε) versus the inverse box side length 1/ε. The red lines represent least-squares fits of straight lines to the data points (black squares), the slopes of which yield D₀ values for (a) grain #1 and (b) grain #2 in Fig. 1(b) and (c) grain #3 in Fig. 1(c).

Figure 3

Frequency histogram of the box-counting fractal dimension D₀ of the grains displayed in Fig. 1(b) and (c). The red histogram bars correspond to the fractal grain morphologies of the nanocrystalline sample following heat treatment (Fig. 1(b)), whereas the blue bars derive from the conventional polycrystalline microstructure prepared by solidification from the melt (Fig. 1(c)).