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Correlative image learning of chemo-mechanics in phase-transforming solids

Abstract

Constitutive laws underlie most physical processes in nature. However, learning such equations in heterogeneous solids (for example, due to phase separation) is challenging. One such relationship is between composition and eigenstrain, which governs the chemo-mechanical expansion in solids. Here we developed a generalizable, physically constrained image-learning framework to algorithmically learn the chemo-mechanical constitutive law at the nanoscale from correlative four-dimensional scanning transmission electron microscopy and X-ray spectro-ptychography images. We demonstrated this approach on LiXFePO4, a technologically relevant battery positive electrode material. We uncovered the functional form of the composition–eigenstrain relation in this two-phase binary solid across the entire composition range (0 ≤ X ≤ 1), including inside the thermodynamically unstable miscibility gap. The learned relation directly validates Vegard’s law of linear response at the nanoscale. Our physics-constrained data-driven approach directly visualizes the residual strain field (by removing the compositional and coherency strain), which is otherwise impossible to quantify. Heterogeneities in the residual strain arise from misfit dislocations and were independently verified by X-ray diffraction line profile analysis. Our work provides the means to simultaneously quantify chemical expansion, coherency strain and dislocations in battery electrodes, which has implications on rate capabilities and lifetime. Broadly, this work also highlights the potential of integrating correlative microscopy and image learning for extracting material properties and physics.

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Fig. 1: Schematic of inverse image-learning framework for constitutive equations.
Fig. 2: Inverse image learning of composition–eigenstrain relation.
Fig. 3: Chemo-mechanical insights of LiXFePO4.
Fig. 4: Dislocation-induced X-ray line broadening.

Data availability

The 4D-STEM and X-ray microscopy data associated with this paper can be found at https://data.matr.io/6/. Additional data are available from the corresponding authors upon reasonable request.

Code availability

The codes used for image registration and image inversion can be accessed at https://github.com/dhtdean/correlative-image-learning. Additional code is available from the corresponding authors upon reasonable request.

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Acknowledgements

This work was supported by the Toyota Research Institute through the Accelerated Materials Design and Discovery programme. X-ray ptychography development was supported by the US Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (contract DE-AC02-76SF00515). This research used resources of the Advanced Light Source, which is a DOE Office of Science User Facility, under contract no. DE-AC02-05CH11231. Work by W.C. was supported by DOE, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award no. DE-SC0010412. Work at the Molecular Foundry was supported by the DOE Office of Science, Office of Basic Energy Sciences under contract no. DE-AC02-05CH11231. Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by DOE, Office of Science, Office of Basic Energy Sciences under contract no. DE-AC02-76SF00515. Part of this work was performed at the Stanford Nano Shared Facilities and Stanford Nano-fabrication Facility, supported by the National Science Foundation under award ECCS-1542152. We thank C. Gopal, P. Herring and A. Anapolsky for assistance in the 4D-STEM data pipeline set-up. We thank N. Nadkarni for insightful discussions on the mechanics that inspired this work. We thank M. Kiani for insightful discussions on dislocations. We thank H. Mohammad and Y. Ye for helpful discussions on PDE-constrained optimization algorithms. We thank H. Thaman and E. Kaeli for manuscript review.

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H.D.D., N.J., W.C.C. and A.M.M. conceived the experiments. H.D.D., N.J. and E.G.L. performed the synthesis and materials characterization. H.D.D. and N.J. performed the STXM and ptychography experiments. H.D.D. performed the STXM and X-ray spectro-ptychography data analysis. Y.-S.Y. and D.A.S. contributed to the scanning transmission X-ray microscopy and ptychography experiments. L.H. performed the 4D-STEM experiments. C.O. performed the image registration. L.H. and B.H.S. performed the 4D-STEM analysis. H.D.D., H.Z. and M.Z.B. developed and performed the inverse image-learning optimization. R.Y. and J.L. contributed to the early algorithmic exploration of PDE-constrained optimization. H.D.D. and W.C. performed the 2D phase-field simulation and dislocation density optimization. D.F. performed the 3D phase-field simulation. H.D.D. performed the residual strain analysis. H.D.D., W.C. and A.B. performed the X-ray line profile analysis. Y.-S.Y. analysed the ptycho-tomography data. H.D.D. prepared the manuscript. All authors contributed to the discussion of the results and writing of the manuscript.

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Correspondence to Andrew M. Minor or William C. Chueh.

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Deng, H.D., Zhao, H., Jin, N. et al. Correlative image learning of chemo-mechanics in phase-transforming solids. Nat. Mater. 21, 547–554 (2022). https://doi.org/10.1038/s41563-021-01191-0

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