A Simple Method for Calculating the Possible Habit Planes Containing One Set of Dislocations and its Applications to fcc
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INTRODUCTION
THE morphologies of the product phases developed from phase transformations are an important aspect of microstructure that controls the mechanical properties of the materials. The habit plane (HP) of a product phase is an essential feature for understanding the crystallographic morphology.[1,2] The HPs in many metallic materials often exhibit irrational orientations. An irrational orientation is an orientation that cannot be indexed with a set of integers, especially small integers. This irrational nature in a precipitation transformation can be rationalized by the crystallographic models based on the geometrical matching in the HP with the assumption that good atomic matching implies a low value of interfacial energy. Since a true invariant plane strain is rare between two dissimilar crystals, usually a planar interface with a sufficiently large area cannot be fully coherent and it contains at least one set of dislocations to accommodate the interfacial misfit. The structural ledge model[3,4] is a pioneering approach for interpreting the irrational HP in the face-centered cubic (fcc)/body-centered cubic (bcc) system. According to this model, the HP locally contains one set of dislocations together with the structural ledges that also play a role in misfit cancelation. If a slight deviation is allowed from the parallelism of the close-packed planes X.-F. GU, formerly Ph.D. Student with the Laboratory of Advanced Materials, School of Materials Science and Engineering, Tsinghua University, Beijing 100084, P.R. China, is now with the Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan. W.-Z. ZHANG, Professor, is with the Laboratory of Advanced Materials, School of Materials Science and Engineering, Tsinghua University. Contact e-mail: [email protected] Manuscript submitted April 13, 2011. Article published online December 7, 2013 METALLURGICAL AND MATERIALS TRANSACTIONS A
in the original model, the optimum interface can contain a true invariant line (IL).[5–7] The concept that the HP contains an IL has been successfully applied to rationalize the TEM observations of irrational orientation relationship (OR) and/or HP generated in various systems from martensitic or precipitation transformations.[7–17] However, the existence of the IL alone only constrains one of three degrees of freedom in the OR. Thus, specific constraints applied in various models yield different results.[9,10,12,18,19] The O-line criterion,[12] based on the O-lattice theory,[20] is a rather general constraint to the HP.[1,21,22] The criterion requires the HP to contain O-lines, i.e., the misfit in the HP should be fully accommodated by only one set of dislocations. It is believed that this dislocation structure corresponds to a singularity (a local minimum cusp) in the interfacial energy, such that adding other types of dislocations due to deviations in either the OR or the interface orientation will cause an increase of the interfacial energy.[23] The O-line criterion is consistent with the interfacial energy c
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