Variance of the Grain Boundary Character Distribution

Abstract

A method for estimating the variance of the grain boundary character distribution from experimental recovery of the two-point correlation functions is derived. The method is illustrated in 2-D to analyze the variance of “susceptible” boundaries in alloy 304 stainless steel. Comparison of these estimates with results obtained directly from OIM measurements is presented. Introduction When the phenomena governing failure in crystalline materials by fracture or crack propagation is intergranular in nature or a mixture of intergranular and transgranular behaviors, the character of the grain boundaries in the region surrounding the point of interest will govern much of the resulting behavior. Although conditions affecting this behavior will also be governed largely by the ductility and temperature under which the fracture occurs, the intergranular interfaces will affect the ability of the material to drive or stop this fracture [1,2]. Studies regarding stress corrosion cracking (SCC) have shown the effect of special boundaries upon the propagation of cracks in experimental samples [3-7]. Boundaries of low misorientation angle along with coincident site lattice (CSL) boundaries of low value inhibited the propagation of cracks in the material whereas boundaries with large misorientation angles were prone to propagate the crack. Accordingly, all references to “susceptible” grain boundaries in this publication will be of large misorientation angle excluding low value CSLs. The connectivity or percolation of grain boundaries of certain character has also proven to be a governing factor in the fracture mechanics of materials [8]. These percolative paths were found to be largely dependent upon percolation “red bonds” which have the ability or inability to impede these percolation paths [7]. Although these two conditions are clearly related in the failure of materials, the difficulty in relating the two is that percolation theory governs the behavior for infinite lattices and failure by fracture often only occurs in a small region; therefore linking the two becomes the probability of the two effects occurring in the same place and at the point of heightened stress. Therefore these two ideas may only be related when the region of interest is large. Research Perspective. The subject of interest is to develop a model which will link the fraction of grain boundaries necessary to develop a connected path of “susceptible” boundaries across a region of interest where the connected length would pose a serious threat to the integrity of the material. Although the probability of finding a cluster of connected length in a specified area has proven difficult because of the nature of percolation, the ability to determine the fraction of “susceptible” boundaries in a given region can be extracted readily. The authors are currently studying the percolative behavior of grain boundaries in alloy 304 stainless steel in order to establish this connection, but this discussion will be presented elsewhere. The focus of this paper will be on the determination of the variance of “susceptible” boundaries as a function of window size by a new method employing a two-point correlation function. The derivation along with experimental procedures and findings for the variance will be discussed. Variance of LA The Microstructure Function ) , ( s x M . In deriving the new method for sampling the variance of the grain boundary character distribution (GBCD), SV (s) , it is illustrative to consider the 2-D function LA(s), which is the line length per unit area of boundaries of local state s. Here local state refers to any selected set of parameters defining local grain boundary character, such as lattice misorientation, boundary inclination, etc. In 2-D measurements the latter is normally inaccessible. Consider the semi-continuous grain boundary network contained in a given field of view . LA(s) for the region can be recovered in two ways; the first of which is by deconstructing the data points from an Orientation Imaging Microscopy (OIM) dataset into discrete line segments, or grain boundary traces. By adding up the length of all the grain boundary traces of state s and dividing by the area of the scan, the true value for LA(s) can be obtained for the region . The second, new method requires allowing the grain boundary traces from the OIM scan to have a finite thickness, t, and then by sampling the region of interest with the vector x . Define a new microstructure function ) , ( s x M where ± = otherwise 0 type of boundary the from 2 within lies if 1 ) , ( s t x t s x M . (1) Integrating over the region of interest, and taking the limit as the thickness goes to zero one obtains the function LA(s) Ω = → 2 ) ( ) , ( ) ( 1 lim ) ( 0 R t A x d x s x M A s L θ (2)