Grain Boundary Curvature in a Model Ni-Based Superalloy
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IN studies of grain growth, the grain boundary (GB) curvature is an essential parameter because it represents the driving force for GB migration. The thermodynamic driving force for normal grain growth is usually expressed as P ¼ 2cS H
½1
where cS is the GB surface energy and H is the surfaceweighted average of the local curvature over all GBs. Experimentally, the curvature H can be obtained from area tangent and linear intercept measurements[1,2,3] and is given as H ¼ pTA I=2
½2
where TA is the number of points of tangency per unit area between a sweeping test line and a curved GB trace on a two-dimensional (2-D) cross section, and I is the mean linear intercept. We note that while this approach uses measurements from planar sections, the values of H obtained represent the average boundary grain curvature in three-dimensions (3-D). Conventionally, H is often replaced by the linear intercept and is expressed in the following form:[4–7] H ¼ j I 1
½3
The curvature parameter j is the factor required to describe the driving force for normal grain growth KAI SONG and MARK AINDOW are with the Institute of Materials Science, University of Connecticut, Storrs, CT 06269-3136, USA; and Materials Science and Engineering Program, Department of Chemical, Materials and Biomolecular Engineering, University of Connecticut, Storrs, CT 06269-3222, USA. Contact e-mail: m.aindow @uconn.edu Manuscript submitted July 21, 2006. METALLURGICAL AND MATERIALS TRANSACTIONS A
relative to the grain size. Substituting j into Eq. [1] yields P ¼ 2cS ðj=IÞ
½4
which is the expression used frequently for modeling grain growth. In most cases, j is assumed to be a constant specific to the system under consideration,[8] with the value of j depending upon the topology of the grain structure. Thus, j = 0 corresponds to completely flat GBs, while j = 4/3 corresponds to spherical grains. For real grain structures, j adopts intermediate values. For example, Patterson and Liu[5] measured the GB curvature in high-purity aluminum and obtained a value of 0.31 for j. this corresponds to a driving force for grain growth approximately 3 times smaller than the value quoted in conventional models. Considering the importance of curvature in grain growth, however, there have been surprisingly few experimental measurements of H and j reported. In an early article, Haroun and Budworth[9] reported curvature measurements from GBs in magnesia, but their measurements involved certain geometrical assumptions. Aside from the work by Patterson and Liu,[5] the only other known measurements of H and j based on the tangent count method are those by Rios and Fonseca[6,7] for an Al-1 mass pct Mn alloy, which contained Al6Mn particles. These latter authors obtained very similar results to Patterson and Liu, although the numerical values quoted differ by a factor of 2 due to differences in the definition for H. All of the previous measurements of H and j have been performed either in pure materials[5,9] or in dilute alloys:[6,7] we are not aware of any similar studi
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