Benjamin L. Hansen and Dr. Brent Adams, Mechanical Engineering
A great amount of variance exists in the properties of any material. The ability to accurately predict these properties allows for designers to find the “best” material in highly constrained design situations. Many properties, such as elasticity and conductivity, are relatively insensitive to grain boundary interfaces and dislocations, and are thus accurately predicted by simple averaging techniques with a first-order description of the microstructure. However, many properties that effect lifetime to failure such as plastic yielding, intergranular stress corrosion cracking, and embrittlement are sensitive to dislocations and grain boundary interfaces. Therefore, second-order descriptions of the microstructures are needed to estimate these properties. It is not just what is in the material, but how it is arranged that is needed.
One method of characterizing the second-order features of a microstructure is to use orientation correlation functions or two-point statistics. To gather the statistics the starting and ending microstructure characteristics (orientation, phase, etc.) are recorded for each vector of a discrete direction and length placed within the microstructure. The distributions that result from this vector set are the two-point statistics.
Work has been done to understand the grain boundary character distribution function, shortrange (across grain boundaries) two-point statistics with useful results [1], and there has been some recent work on creating structures from orientation correlation functions with a simulated annealing monte carlo method [2]. It is also of interest to control defect sensitive properties with statistical measures and to create three-dimensional structures from only two-dimensional statistical information. Generally, only two-dimensional information can be acquired, unless serial sectioning is preformed on the sample, which is a very long and tedious work.
The objective of this project was to computer generate microstructures solely from two-point statistics in order to study the control of orientation correlation functions over connectivity and topology of grain boundary networks and to determine if three-dimensional models can be created accurately from the two-dimensional statistical information of these functions. Three methods were used to create structures: simulated annealing was used both with a variable metric and a monte carlo method and a new “branching” combinatorics method. In all cases the structures were considered periodic similar to Yeong and Torquato [2], and the statistics were measured from a real microstructure for comparison and to guarantee the statistics existence in a structure. Although the results for the monte carlo method were reproduced, no significant differences were found from the results of Yeong and Torquato, so the reader is referenced to them for the results of the simulated annealing method.
The new branching method differs from the simulated annealing methods used in that it will create structures with the exact statistics given and not approximate the statistics. The branching method uses the number of occurrences generated from statistics to turn the situation into a combinatorics problem. This does create a difficulty that the statistics must be exact to create whole numbers for occurances. The branching method uses a trial and error placement of points to solve for a structure that will recreate the input statistics.
The most intriguing result of using the branching method is the return of exact translated replicates of the original structure. The translation is due to the periodic structure assumption. In cases where only two orientations are involved, there is more than one structure found which matches the statistics. In these cases there is also a mirror structure which matches the statistics. See figure 1. Cases with only two orientations or phases were the only observed cases where multiple structures had the same statistics.
Because the branching method produces exact replicates the connectivity and topology of structures is preserved in the recreation of the structure. Although the simulated annealing methods produce structures with less than 1% error, they are not exact replicates and do not conserve the connectivity or topology of the structures [2]. It was also noted that the branching method is superior to simulated annealing methods when many orientations are involved, as with polycrystals.
Real electron microscopy scans have been reproduced with the translational shift using the branching method. Three dimensional models are possible, but only with the exact statistics that could be gathered by serial sectioning. Since it has been observed that a slight deviation from the exact statistics produce widely variant structures in simulated annealing, it is not possible to produce exact models in three-dimensions from two-dimensional information. Models could be created, but the defect sensitive properties would not be conserved. Despite this, the method still has applications to create structures with pre-defined properties.
References
- Adams, B.L. et al. Description of Orientation Coherence in Polycrystalline Materials. Acta Metal. Vol. 35, No. 12, pp 2935-2946, 1987
- Yeong, C.L.Y. and S. Torquato. Reconstructing Random Media. Physical Review E. Vol. 51, No. 1, pp. 495-506, 1998
Acknowledgements: Appreciation for the opportunity and funding provided by ORCA and the assistance of coworkers’ M. Lyon, A. Henrie, and B. Henrie is acknowledge for this research.