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

Nature Materials volume 21, pages 547–554 (2022)Cite this article

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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.

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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.

References

  1. Sheng, S. & Tu, Z. C. Constitutive relation for nonlinear response and universality of efficiency at maximum power for tight-coupling heat engines. Phys. Rev. E 91, 022136 (2015).

    Article  CAS  Google Scholar 

  2. Jackson, J. D. Classical Electrodynamics 3rd edn (Wiley, 1999).

  3. Magnenet, V., Schiavi-Tritz, J., Huselstein, C. & Rahouadj, R. Constitutive equations for Ca2+-alginate gels. J. Mech. Behav. Biomed. Mater. 5, 90–98 (2012).

    Article  CAS  Google Scholar 

  4. Jop, P., Forterre, Y. & Pouliquen, O. A constitutive law for dense granular flows. Nature 441, 727–730 (2006).

    Article  CAS  Google Scholar 

  5. Fish, J., Wagner, G. J. & Keten, S. Mesoscopic and multiscale modelling in materials. Nat. Mater. 20, 774–786 (2021).

    Article  CAS  Google Scholar 

  6. Vegard, L. Die konstitution der mischkristalle und die raumfüllung der atome. Z. Phys. 5, 17–26 (1921).

    Article  CAS  Google Scholar 

  7. Denton, A. R. & Ashcroft, N. W. Vegard’s law. Phys. Rev. A 43, 3161–3164 (1991).

    Article  CAS  Google Scholar 

  8. Tuller, H. L. & Bishop, S. R. Point defects in oxides: tailoring materials through defect engineering. Annu. Rev. Mater. Res. 41, 369–398 (2011).

    Article  CAS  Google Scholar 

  9. Koerver, R. et al. Chemo-mechanical expansion of lithium electrode materials – on the route to mechanically optimized all-solid-state batteries. Energy Environ. Sci. 11, 2142–2158 (2018).

    Article  CAS  Google Scholar 

  10. Xia, X. et al. Electrochemically reconfigurable architected materials. Nature 573, 205–213 (2019).

    Article  CAS  Google Scholar 

  11. Bishop, S. R. et al. Electro-chemo-mechanics of Solids (Springer, 2017).

  12. Woodford, W. H., Chiang, Y.-M. & Carter, W. C. “Electrochemical shock” of intercalation electrodes: a fracture mechanics analysis. J. Electrochem. Soc. 157, A1052–A1059 (2010).

    Article  CAS  Google Scholar 

  13. Cogswell, D. A. & Bazant, M. Z. Coherency strain and the kinetics of phase separation in LiFePO4 nanoparticles. ACS Nano 6, 2215–2225 (2012).

    Article  CAS  Google Scholar 

  14. Christensen, J. & Newman, J. Stress generation and fracture in lithium insertion materials. J. Solid State Electrochem. 10, 293–319 (2006).

    Article  CAS  Google Scholar 

  15. Baldi, A., Narayan, T. C., Koh, A. L. & Dionne, J. A. In situ detection of hydrogen-induced phase transitions in individual palladium nanocrystals. Nat. Mater. 13, 1143–1148 (2014).

    Article  CAS  Google Scholar 

  16. Wagemaker, M. et al. Dynamic solubility limits in nanosized olivine LiFePO4. J. Am. Chem. Soc. 133, 10222–10228 (2011).

    Article  CAS  Google Scholar 

  17. Yamada, A. et al. Room-temperature miscibility gap in LixFePO4. Nat. Mater. 5, 357–360 (2006).

    Article  CAS  Google Scholar 

  18. Meethong, N., Huang, H.-Y. S., Carter, W. C. & Chiang, Y.-M. Size-dependent lithium miscibility gap in nanoscale Li1-xFePO4. Electrochem. Solid State Lett. 10, A134–138 (2007).

    Article  CAS  Google Scholar 

  19. Stillinger, F. H. Exponential multiplicity of inherent structures. Phys. Rev. E 59, 48–51 (1999).

    Article  CAS  Google Scholar 

  20. Shapiro, D. A. et al. Chemical composition mapping with nanometre resolution by soft X-ray microscopy. Nat. Photon. 8, 765–769 (2014).

    Article  CAS  Google Scholar 

  21. Ophus, C. Four-dimensional scanning transmission electron microscopy (4D-STEM): from scanning nanodiffraction to ptychography and beyond. Microsc. Microanal. 25, 563–582 (2019).

    Article  CAS  Google Scholar 

  22. Lim, J. et al. Origin and hysteresis of lithium compositional spatiodynamics within battery primary particles. Science 353, 566–571 (2016).

    Article  CAS  Google Scholar 

  23. Ulvestad, A. et al. Topological defect dynamics in operando battery nanoparticles. Science 348, 1344–1347 (2015).

    Article  CAS  Google Scholar 

  24. Liu, D. et al. Demonstration of a novel focusing small-angle neutron scattering instrument equipped with axisymmetric mirrors. Nat. Commun. 4, 2556 (2013).

    Article  CAS  Google Scholar 

  25. Panova, O. et al. Diffraction imaging of nanocrystalline structures in organic semiconductor molecular thin films. Nat. Mater. 18, 860–865 (2019).

    Article  CAS  Google Scholar 

  26. Xu, K., Huang, D. Z. & Darve, E. Learning constitutive relations using symmetric positive definite neural networks. J. Comput. Phys. 428, 110072 (2021).

    Article  Google Scholar 

  27. Kalinin, S. V., Sumpter, B. G. & Archibald, R. K. Big–deep–smart data in imaging for guiding materials design. Nat. Mater. 14, 973–980 (2015).

    Article  CAS  Google Scholar 

  28. Zhao, H., Braatz, R. D. & Bazant, M. Z. Image inversion and uncertainty quantification for constitutive laws of pattern formation. J. Comput. Phys. 436, 110279 (2021).

    Article  Google Scholar 

  29. Zhao, H., Storey, B. D., Braatz, R. D. & Bazant, M. Z. Learning the physics of pattern formation from images. Phys. Rev. Lett. 124, 60201 (2020).

    Article  CAS  Google Scholar 

  30. Seemann, R., Herminghaus, S. & Jacobs, K. Dewetting patterns and molecular forces: a reconciliation. Phys. Rev. Lett. 86, 5534–5537 (2001).

    Article  CAS  Google Scholar 

  31. Morozovska, A. N., Eliseev, E. A., Chen, D., Nelson, C. T. & Kalinin, S. V. Building a free-energy functional from atomically resolved imaging: atomic-scale phenomena in La-doped BiFeO3. Phys. Rev. B 99, 195440 (2019).

    Article  CAS  Google Scholar 

  32. Park, J. et al. Fictitious phase separation in Li layered oxides driven by electro-autocatalysis. Nat. Mater. 20, 991–999 (2021).

  33. Nelson, C. T. et al. Exploring physics of ferroelectric domain walls via Bayesian analysis of atomically resolved STEM data. Nat. Commun. 11, 6361 (2020).

    Article  CAS  Google Scholar 

  34. Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, J. N. Data-driven discovery of partial differential equations. Sci. Adv. 3, 1602614 (2017).

    Article  Google Scholar 

  35. Tarantola, A. Inverse Problem Theory and Methods for Model Parameter Estimation (SIAM, 2005).

  36. Tang, M., Carter, W. C. & Chiang, Y.-M. Electrochemically driven phase transitions in insertion electrodes for lithium-ion batteries: examples in lithium metal phosphate olivines. Annu. Rev. Mater. Res. 40, 501–529 (2010).

    Article  CAS  Google Scholar 

  37. Chen, G., Song, X. & Richardson, T. J. Electron microscopy study of the LiFePO4 to FePO4 phase transition. Electrochem. Solid State Lett. 9, A295–A298 (2006).

    Article  CAS  Google Scholar 

  38. Padhi, A. K., Nanjundaswamy, K. S. & Goodenough, J. B. Phospho-olivines as positive-electrode materials for rechargeable lithium batteries. J. Electrochem. Soc. 144, 1188–1194 (1997).

    Article  CAS  Google Scholar 

  39. Nadkarni, N. et al. Interplay of phase boundary anisotropy and electro-auto-catalytic surface reactions on the lithium intercalation dynamics in LiXFePO4 platelet like nanoparticles. Phys. Rev. Mater. 2, 085406 (2018).

    Article  CAS  Google Scholar 

  40. Thibault, P. et al. High-resolution scanning X-ray diffraction microscopy. Science 321, 379–382 (2008).

    Article  CAS  Google Scholar 

  41. Savitzky, B. H. et al. py4DSTEM: A software package for four-dimensional scanning transmission electron microscopy data analysis. Microsc. Microanal. 27, 712–743 (2021).

    Article  CAS  Google Scholar 

  42. Borbély, A. & Groma, I. Variance method for the evaluation of particle size and dislocation density from X-ray Bragg peaks. Appl. Phys. Lett. 79, 1772–1774 (2001).

    Article  CAS  Google Scholar 

  43. Cheng, Y.-T. & Verbrugge, M. W. Diffusion-induced stress, interfacial charge transfer, and criteria for avoiding crack initiation of electrode particles. J. Electrochem. Soc. 157, A508–A516 (2010).

    Article  CAS  Google Scholar 

  44. Hughes, L. A. et al. Correlative analysis of structure and chemistry of LixFePO4 platelets using 4D-STEM and X-ray ptychography. Mater. Today https://doi.org/10.1016/j.mattod.2021.10.031 (2021).

  45. Li, Y. et al. Fluid-enhanced surface diffusion controls intraparticle phase transformations. Nat. Mater. 17, 915–922 (2018).

    Article  CAS  Google Scholar 

  46. Kobayashi, S., Kuwabara, A., Fisher, C. A. J., Ukyo, Y. & Ikuhara, Y. Microscopic mechanism of biphasic interface relaxation in lithium iron phosphate after delithiation. Nat. Commun. 9, 2863 (2018).

    Article  CAS  Google Scholar 

  47. Laffont, L. et al. Study of the LiFePO4/FePO4 two-phase system by high-resolution electron energy loss spectroscopy. Chem. Mater. 18, 5520–5529 (2006).

    Article  CAS  Google Scholar 

  48. Tang, M., Belak, J. F. & Dorr, M. R. Anisotropic phase boundary morphology in nanoscale olivine electrode particles. J. Phys. Chem. C 115, 4922–4926 (2011).

    Article  CAS  Google Scholar 

  49. Mura, T. Micromechanics of Defects in Solids (Springer Science & Business Media, 2013).

  50. Egerton, R. F. Physical Principles of Electron Microscopy (Springer, 2005).

  51. Qin, X. et al. Hydrothermally synthesized LiFePO4 crystals with enhanced electrochemical properties: simultaneous suppression of crystal growth along [010] and antisite defect formation. Phys. Chem. Chem. Phys. 14, 2669–2677 (2012).

    Article  CAS  Google Scholar 

  52. Chen, J. & Graetz, J. Study of antisite defects in hydrothermally prepared LiFePO4 by in situ X-ray diffraction. ACS Appl. Mater. Interfaces 3, 1380–1384 (2011).

    Article  CAS  Google Scholar 

  53. Li, Y. et al. Current-induced transition from particle-by-particle to concurrent intercalation in phase-separating battery electrodes. Nat. Mater. 13, 1149–1156 (2014).

    Article  CAS  Google Scholar 

  54. Evangelidis, G. D. & Psarakis, E. Z. Parametric image alignment using enhanced correlation coefficient maximization. IEEE Trans. Pattern Anal. Mach. Intell. 30, 1858–1865 (2008).

    Article  Google Scholar 

  55. Farmand, M. et al. Near-edge X-ray refraction fine structure microscopy. Appl. Phys. Lett. 110, 063101 (2017).

    Article  CAS  Google Scholar 

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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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Authors and Affiliations

  1. Department of Materials Science and Engineering, Stanford University, Stanford, CA, USA

    Haitao D. Deng, Norman Jin, Eder G. Lomeli & William C. Chueh

  2. Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA

    Haitao D. Deng, Rui Yan & Jueyi Liu

  3. Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

    Hongbo Zhao, Dimitrios Fraggedakis & Martin Z. Bazant

  4. National Center for Electron Microscopy, Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Lauren Hughes, Benjamin H. Savitzky, Colin Ophus & Andrew M. Minor

  5. Centre SMS, Georges Friedel Laboratory (UMR 5307), Mines Saint-Etienne, Univ. Lyon, CNRS, Saint-Etienne, France

    András Borbély

  6. Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

    Young-Sang Yu & David A. Shapiro

  7. Department of Physics, Chungbuk National University, Cheongju, Republic of Korea

    Young-Sang Yu

  8. Department of Mechanical Engineering, Stanford University, Stanford, CA, USA

    Wei Cai

  9. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Martin Z. Bazant

  10. Department of Materials Science and Engineering, University of California, Berkeley, CA, USA

    Andrew M. Minor

  11. Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Stanford University, Menlo Park, CA, USA

    William C. Chueh

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Contributions

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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Supplementary Figs. 1–15, Tables 1–5 and Notes 1–6.

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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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