# Grain boundary structure–property model inference using polycrystals: the overdetermined case

- 160 Downloads

## Abstract

Efforts to construct predictive grain boundary (GB) structure–property models have historically relied on property measurements or calculations made on *bicrystals*. Experimental bicrystals can be difficult or expensive to fabricate, and computational constraints limit atomistic bicrystal simulations to high-symmetry GBs (i.e., those with small enough GB periodicity). Although the use of bicrystal property data to construct GB structure–property models is more direct, in many experimental situations the only type of data available may be measurements of the effective properties of polycrystals. In this work, we investigate the possibility of inferring GB structure–property models from measurements of the homogenized effective properties of polycrystals *when the form of the structure–property model is unknown*. We present an idealized case study in which GB structure–property models for diffusivity are inferred from noisy simulation results of two-dimensional microstructures, under the assumption that the number of polycrystal measurements available is larger than the number of parameters in the inferred model. We also demonstrate how uncertainty quantification for the inferred structure–property models is easily performed within this framework.

## Notes

### Acknowledgements

The material presented here is based upon work supported by the National Science Foundation under Grant No. 1610077. We thank Jarrod M. Lund and Tyler R. Critchfield for their assistance in developing the Monte Carlo code to assign grain orientations for the two-dimensional polycrystal templates. We would also like to show our gratitude for the guidance and insights from David Page and Akash Amalaraj during the course of this research.

## References

- 1.Kaur I, Mishin Y, Gust W (1995) Fundamentals of grain and interphase boundary diffusion, 3rd edn. Willey, HobokenGoogle Scholar
- 2.Hugo RC, Hoagland RG (2000) The kinetics of gallium penetration into aluminum grain boundaries—in situ TEM observations and atomistic models. Acta Mater 48(8):1949–1957. https://doi.org/10.1016/S1359-6454(99)00463-2 CrossRefGoogle Scholar
- 3.Olmsted DL, Foiles SM, Holm EA (2009) Survey of computed grain boundary properties in face-centered cubic metals: I. Grain boundary energy. Acta Mater 57(13):3694–3703. https://doi.org/10.1016/j.actamat.2009.04.007 CrossRefGoogle Scholar
- 4.Olmsted DL, Holm EA, Foiles SM (2009) Survey of computed grain boundary properties in face-centered cubic metals-II: grain boundary mobility. Acta Mater 57(13):3704–3713. https://doi.org/10.1016/j.actamat.2009.04.015 CrossRefGoogle Scholar
- 5.Rollett AD, Yang CC, Mullins WW, Adams BL, Wu CT, Kinderlehrer D, Ta’asan S, Manolache F, Liu C, Livshits I, Mason D, Talukder A, Ozdemir S, Casasent D, Morawiec A, Saylor D, Rohrer GS, Demirel M, El-Dasher B, Yang W (2001) Grain boundary property determination through measurement of triple junction geometry and crystallography. Recryst Grain 1(2):165–175Google Scholar
- 6.Read WT, Shockley W (1950) Dislocation models of crystal grain boundaries. Phys Rev 78(3):275–289CrossRefGoogle Scholar
- 7.Bulatov VV, Reed BW, Kumar M (2014) Grain boundary energy function for FCC metals. Acta Mater 65:161–175. https://doi.org/10.1016/j.actamat.2013.10.057 CrossRefGoogle Scholar
- 8.Binci M, Fullwood D, Kalidindi SR (2008) A new spectral framework for establishing localization relationships for elastic behavior of composites and their calibration to finite-element models. Acta Mater 56(10):2272–2282. https://doi.org/10.1016/j.actamat.2008.01.017 CrossRefGoogle Scholar
- 9.Fast T, Kalidindi SR (2011) Formulation and calibration of higher-order elastic localization relationships using the MKS approach. Acta Mater 59(11):4595–4605. https://doi.org/10.1016/j.actamat.2011.04.005 CrossRefGoogle Scholar
- 10.Fullwood DT, Kalidindi SR, Adams BL, Ahmadi S (2009) A discrete fourier transform framework for localization relations. Comput Mater Continua 9(1):25–39. https://doi.org/10.3970/cmc.2009.009.025 CrossRefGoogle Scholar
- 11.Kalidindi SR, Landi G, Fullwood DT (2008) Spectral representation of higher-order localization relationships for elastic behavior of polycrystalline cubic materials. Acta Mater 56(15):3843–3853. https://doi.org/10.1016/j.actamat.2008.01.058 CrossRefGoogle Scholar
- 12.Landi G, Kalidindi SR (2010) Thermo-elastic localization relationships for multi-phase composites. Comput Mater Continua 16(3):273–293. https://doi.org/10.3970/cmc.2010.016.273 CrossRefGoogle Scholar
- 13.Yabansu YC, Patel DK, Kalidindi SR (2014) Calibrated localization relationships for elastic response of polycrystalline aggregates. Acta Mater 81:151–160. https://doi.org/10.1016/j.actamat.2014.08.022 CrossRefGoogle Scholar
- 14.Yabansu YC, Kalidindi SR (2015) Representation and calibration of elastic localization kernels for a broad class of cubic polycrystals. Acta Mater 94:26–35. https://doi.org/10.1016/j.actamat.2015.04.049 CrossRefGoogle Scholar
- 15.Li DY, Szpunar JA (1992) Determination of single crystals’ elastic constants from the measurement of ultrasonic velocity in the polycrystalline material. Acta Metall Mater 40(12):3277–3283. https://doi.org/10.1016/0956-7151(92)90041-C CrossRefGoogle Scholar
- 16.Hayakawa M, Imai S, Oka M (1985) Determination of single-crystal elastic constants from a cubic polycrystalline aggregate. J Appl Crystallogr 18:513–518. https://doi.org/10.1107/S0021889885010809 CrossRefGoogle Scholar
- 17.Patel DK, Al-Harbi HF, Kalidindi SR (2014) Extracting single-crystal elastic constants from polycrystalline samples using spherical nanoindentation and orientation measurements. Acta Mater 79:108–116. https://doi.org/10.1016/j.actamat.2014.07.021 CrossRefGoogle Scholar
- 18.Gold L (1950) Evaluation of the stiffness coefficients for beryllium from ultrasonic measurements in polycrystalline and single crystal specimens. Phys Rev 77(3):390–395. https://doi.org/10.1103/PhysRev.77.390 CrossRefGoogle Scholar
- 19.Wright SI (1994) Estimation of single-crystal elastic constants from textured polycrystal measurements. J Appl Crystallogr 27(5):794–801. https://doi.org/10.1107/S0021889894001883 CrossRefGoogle Scholar
- 20.Gnäupel-Herold T, Brand PC, Prask HJ (1998) Calculation of single-crystal elastic constants for cubic crystal symmetry from powder diffraction data. J Appl Crystallogr 31(6):929–935. https://doi.org/10.1107/S002188989800898X CrossRefGoogle Scholar
- 21.Haldipur P, Margetan FJ, Thompson RB (2004) Estimation of single-crystal elastic constants from ultrasonic measurements on polycrystalline specimens. AIP Conf Proc 700(2004):1061–1068. https://doi.org/10.1063/1.1711735 CrossRefGoogle Scholar
- 22.Haldipur P, Margetan FJ, Thompson RB (2006) Estimation of single-crystal elastic constants of polycrystalline materials from back-scattered grain noise. AIP Conf Proc 820(II(2006)):1133–1140. https://doi.org/10.1063/1.2184652 CrossRefGoogle Scholar
- 23.Gasteau D, Chigarev N, Ducousso-Ganjehi L, Gusev VE, Jenson F, Calmon P, Tournat V (2016) Single crystal elastic constants evaluated with surface acoustic waves generated and detected by lasers within polycrystalline steel samples. J Appl Phys 119(4):043103. https://doi.org/10.1063/1.4940367 CrossRefGoogle Scholar
- 24.Du X, Zhao JC (2017) Facile measurement of single-crystal elastic constants from polycrystalline samples. npj Comput Mater 3(1):1–7. https://doi.org/10.1038/s41524-017-0019-x CrossRefGoogle Scholar
- 25.Sha G (2018) A simultaneous non-destructive characterisation method for grain size and single-crystal elastic constants of cubic polycrystals from ultrasonic measurements. Insight: Non-Destr Test Cond Monit 60(4):190–193. https://doi.org/10.1784/insi.2018.60.4.190 CrossRefGoogle Scholar
- 26.Johnson OK, Li L, Demkowicz MJ, Schuh CA (2015) Inferring grain boundary structure-property relations from effective property measurements. J Mater Sci 50(21):6907–6919. https://doi.org/10.1007/s10853-015-9241-4 CrossRefGoogle Scholar
- 27.Johnson OK, Lund JM, Critchfield TR (2018) Spectral graph theory for characterization and homogenization of grain boundary networks. Acta Mater 146:42–54. https://doi.org/10.1016/j.actamat.2017.11.054 CrossRefGoogle Scholar
- 28.Lazar EA, MacPherson RD, Srolovitz DJ (2009) A more accurate two-dimensional grain growth algorithm. Acta Mater 58:364–372. https://doi.org/10.1016/j.actamat.2009.09.008 CrossRefGoogle Scholar
- 29.Frary M, Schuh CA (2005) Grain boundary networks: scaling laws, preferred cluster structure, and their implications for grain boundary engineering. Acta Mater 53(16):4323–4335. https://doi.org/10.1016/j.actamat.2005.05.030 CrossRefGoogle Scholar
- 30.Gertsman VY, Tangri K (1995) Computer simulation study of grain boundary and triple junction distributions in microstructures formed by multiple twinning. Acta Metall Mater 43(6):2317–2324CrossRefGoogle Scholar
- 31.Johnson OK, Schuh CA (2013) The uncorrelated triple junction distribution function: towards grain boundary network design. Acta Mater 61(8):2863–2873. https://doi.org/10.1016/j.actamat.2013.01.025 CrossRefGoogle Scholar
- 32.Fortier P, Miller WA, Aust KT (1997) Triple junction and grain boundary character distributions in metallic materials. Acta Mater 45(8):4–9CrossRefGoogle Scholar
- 33.Yi YS, Kim JS (2004) Characterization methods of grain boundary and triple junction distributions. Scripta Mater 50(6):855–859. https://doi.org/10.1016/j.scriptamat.2003.12.010 CrossRefGoogle Scholar
- 34.Davies P, Randle V, Watkins G, Davies H (2002) Triple junction distribution profiles as assessed by electron backscatter diffraction. J Mater Sci 37(19):4203–4209. https://doi.org/10.1023/A:1020052306493 CrossRefGoogle Scholar
- 35.Rollett AD, Barmak K (2014) Orientation mapping. In: Laughlin DE, Hono K (eds) Physical metallurgy. Elsevier, Amsterdam, pp 1113–1141. https://doi.org/10.1016/B978-0-444-53770-6.00011-3 CrossRefGoogle Scholar
- 36.Brough I, Bate PS, Humphreys FJ (2006) Optimising the angular resolution of EBSD. Mater Sci Technol 22(11):1279–1286. https://doi.org/10.1179/174328406X130902 CrossRefGoogle Scholar
- 37.Zaefferer S (2007) On the formation mechanisms, spatial resolution and intensity of backscatter Kikuchi patterns. Ultramicroscopy 107(2–3):254–266. https://doi.org/10.1016/j.ultramic.2006.08.007 CrossRefGoogle Scholar
- 38.Kacher J, Landon C, Adams BL, Fullwood D (2009) Bragg’s Law diffraction simulations for electron backscatter diffraction analysis. Ultramicroscopy 109(9):1148–1156. https://doi.org/10.1016/j.ultramic.2009.04.007 CrossRefGoogle Scholar
- 39.Wilkinson AJ, Meaden G, Dingley DJ (2006) High-resolution elastic strain measurement from electron backscatter diffraction patterns: New levels of sensitivity. Ultramicroscopy 106(4–5):307–313. https://doi.org/10.1016/j.ultramic.2005.10.001 CrossRefGoogle Scholar
- 40.Chen Y, Schuh CA (2006) Diffusion on grain boundary networks: percolation theory and effective medium approximations. Acta Mater 54(18):4709–4720. https://doi.org/10.1016/j.actamat.2006.06.011 CrossRefGoogle Scholar
- 41.Moghadam MM, Rickman JM, Harmer MP, Chan HM (2015) The role of boundary variability in polycrystalline grain-boundary diffusion. J Appl Phys 117(4):045311. https://doi.org/10.1063/1.4906778 CrossRefGoogle Scholar
- 42.Tarantola A, Valette B (1982) Inverse problems = quest for information. J Geophys 50(3):159–170. https://doi.org/10.1038/nrn1011 CrossRefGoogle Scholar
- 43.Mosegaard K, Tarantola A (2002) Probabilistic approach to inverse problems. In: Lee WH, Jennings P, Kisslinger C, Kanamori H (eds) International handbook of earthquake and engineering seismology (Part A). Academic Press, Cambridge, pp 237–265. https://doi.org/10.1016/S0074-6142(02)80219-4 CrossRefGoogle Scholar
- 44.Tarantola A (2005) Inverse problem theory and methods for model parameter estimation. Society for Industrial and Applied Mathematics, PhiladelphiaCrossRefGoogle Scholar
- 45.Aggarwal R, Demkowicz MJ, Marzouk YM (2015) Bayesian inference of substrate properties from film behavior. Modell Simul Mater Sci Eng 23(1):15. https://doi.org/10.1088/0965-0393/23/1/015009 CrossRefGoogle Scholar
- 46.Aggarwal R, Demkowicz MJ, Marzouk YM (2016) Information-driven experimental design in materials science. In: Lookman T, Alexander FJ, Rajan K (eds) Information science for materials discovery and design, vol 225. Springer, Berlin, pp 13–44. https://doi.org/10.1007/978-3-319-23871-5 CrossRefGoogle Scholar
- 47.Johnson OK, Schuh CA (2018) Texture mediated grain boundary network design in three dimensions. Mech Mater 118:94–105. https://doi.org/10.1016/j.mechmat.2017.12.001 CrossRefGoogle Scholar
- 48.Biscondi M (1984) Intergranular diffusion and grain-boundary structure. In: Lacombe P (ed) Physical chemistry of the solid state: applications to metals and their compounds. Elsevier Science Publishers B.V, Amsterdam, pp 225–239Google Scholar
- 49.Sommer J, Herzig C, Mayer S, Gust W (1989) Grain boundary self-diffusion in silver bicrystals. Defect Diffus Forum 66–69:843–848Google Scholar
- 50.Quey R, Dawson PR, Barbe F (2011) Large-scale 3D random polycrystals for the finite element method: generation, meshing and remeshing. Comput Methods Appl Mech Eng 200(17–20):1729–1745. https://doi.org/10.1016/j.cma.2011.01.002 CrossRefGoogle Scholar
- 51.Quey R (2017) Neper Reference Manual; The documentation for Neper 3.0.2; A software package for polycrystal generation and meshing. http://neper.sourceforge.net/
- 52.Quey R, Renversade L (2018) Optimal polyhedral description of 3D polycrystals: method and application to statistical and synchrotron X-ray diffraction data. Comput Methods Appl Mech Eng 330:308–333. https://doi.org/10.1016/j.cma.2017.10.029 CrossRefGoogle Scholar
- 53.Quey R, Polycrystal Generation and Meshing | Neper 3.0.2. http://neper.sourceforge.net/
- 54.Weinzierl S (2000) Introduction to Monte Carlo methods. arXiv:0006269
- 55.Kolmogorov AN (1956) Foundations of the theory of probability, Second English edn. Chelsea Publishing Company, New York English translation edited by Nathan MorrisonGoogle Scholar