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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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.
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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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.
The authors declare no competing interests.
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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