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Hidden diversity of vacancy networks in Prussian blue analogues


Prussian blue analogues (PBAs) are a diverse family of microporous inorganic solids, known for their gas storage ability1, metal-ion immobilization2, proton conduction3, and stimuli-dependent magnetic4,5, electronic6 and optical7 properties. This family of materials includes the double-metal cyanide catalysts8,9 and the hexacyanoferrate/hexacyanomanganate battery materials10,11. Central to the various physical properties of PBAs is their ability to reversibly transport mass, a process enabled by structural vacancies. Conventionally presumed to be random12,13, vacancy arrangements are crucial because they control micropore-network characteristics, and hence the diffusivity and adsorption profiles14,15. The long-standing obstacle to characterizing the vacancy networks of PBAs is the inaccessibility of single crystals16. Here we report the growth of single crystals of various PBAs and the measurement and interpretation of their X-ray diffuse scattering patterns. We identify a diversity of non-random vacancy arrangements that is hidden from conventional crystallographic powder analysis. Moreover, we explain this unexpected phase complexity in terms of a simple microscopic model that is based on local rules of electroneutrality and centrosymmetry. The hidden phase boundaries that emerge demarcate vacancy-network polymorphs with very different micropore characteristics. Our results establish a foundation for correlated defect engineering in PBAs as a means of controlling storage capacity, anisotropy and transport efficiency.

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Fig. 1: Structure of PBAs.
Fig. 2: Single-crystal diffuse scattering from PBAs.
Fig. 3: Vacancy network phase diagram.
Fig. 4: Statistical properties of micropore networks.

Data availability

The raw data on which this manuscript is based are openly available for download from These data include the scattering images given in Fig. 2 and the Monte Carlo configurations from which Figs. 3 and 4 are derived.

Code availability

All custom code used in this study was developed using widely available algorithms. Copies of the code used can be obtained upon request.


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A.S. and A.L.G. gratefully acknowledge financial support from the Leverhulme Trust UK (grant no. RPG-2015-292), and T.D.B. acknowledges F.W.O.–Vlaanderen (Research Foundation Flanders) for a Postdoctoral Fellowship. A.S. thanks the Swiss National Science Foundation for Ambizione and PostDoc Mobility Fellowships (PZ00P2_180035, P2EZP2_155608) and Diamond Light Source for the provision of beamtime (MT13639, MT20876, CY22610). M.L.R.G. thanks the Consejo Nacional de Ciencia y Tecnología (Mexico) for a scholarship. A.L.G. thanks the European Research Council for funding (grant nos 279705 and 788144), P. D. Battle (Oxford) and A. R. Overy (Oxford) for valuable discussions, N. P. Funnell (ISIS), J. A. Hill (UCL) and C. S. Coates (Oxford) for assistance with single-crystal growth, and A. L. Thompson for assistance with synchrotron measurements.

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Authors and Affiliations



A.S., T.D.B., H.-B.B. and A.L.G. designed the research. H.L.B.B., M.L.R.G., H.J.G. and T.D.B. synthesized the materials. A.S., D.C., A.B. and H.-B.B. measured the single-crystal diffuse scattering patterns. A.S. performed the 3D-ΔPDF refinement. A.S. and A.L.G. developed and implemented the Monte Carlo model. A.S. and T.D.B. calculated percolation properties. A.S. and A.L.G. wrote the manuscript, with input from all authors.

Corresponding author

Correspondence to Andrew L. Goodwin.

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The authors declare no competing interests.

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Peer review information Nature thanks Simon Billinge, T. Richard Welberry and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Representative pore networks.

Representative pore networks for each phase within the Monte Carlo simulated phase diagram.

Extended Data Fig. 2 Representative 3D-ΔPDF.

a, Experimental diffuse scattering from the Co[Co] sample, hk0 section; voxel size is 1/30 reciprocal lattice units. b, Experimental and model 3D-ΔPDF map of the Co[Co] sample, uv0 section.

Extended Data Fig. 3 M′-site models.

The structure of the [Co(CN)6]3− and ‘vacancy’ moieties used in our Co[Co] model.

Extended Data Fig. 4 Alternative diffuse scattering phase map.

Diffuse scattering calculated with the modified model Hamiltonian; equation (2).

Extended Data Fig. 5 Diffuse scattering tiles.

Comparison of projected experimental diffuse scattering with the model diffuse scattering tiles for our various PBA samples; compare with Fig. 3.

Extended Data Fig. 6 Satellite reflections.

Satellite reflections in our Cu[Co] sample. The inset shows one specific satellite at (7.43, 1, 0).

Extended Data Fig. 7 Inelastic scattering.

Diffuse scattering in the Mn[Co]′ sample. Note that the intensity of the diffuse scattering at the (h + 1/3, k + 1/3, l) positions increases with increasing l; the scattering in the hk4 layer (right) is stronger than in the hk2 layer (left).

Extended Data Table 1 Synthesis summary
Extended Data Table 2 Data collection strategies
Extended Data Table 3 3D-ΔPDF model

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Simonov, A., De Baerdemaeker, T., Boström, H.L. et al. Hidden diversity of vacancy networks in Prussian blue analogues. Nature 578, 256–260 (2020).

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