Abstract
Topological defects—extended lattice deformations that are robust against local defects and annealing—have been exploited to engineer novel properties in both hard and soft materials. Yet, their formation kinetics and nanoscale three-dimensional structure are poorly understood, impeding their benefits for nanofabrication. We describe the fabrication of a pair of topological defects in the volume of a single-diamond network (space group Fd \(\bar{3}\)m) templated into gold from a triblock terpolymer crystal. Using X-ray nanotomography, we resolve the three-dimensional structure of nearly 70,000 individual single-diamond unit cells with a spatial resolution of 11.2 nm, allowing analysis of the long-range order of the network. The defects observed morphologically resemble the comet and trefoil patterns of equal and opposite half-integer topological charges observed in liquid crystals. Yet our analysis of strain in the network suggests typical hard matter behaviour. Our analysis approach does not require a priori knowledge of the expected positions of the nodes in three-dimensional nanostructured systems, allowing the identification of distorted morphologies and defects in large samples.
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Data availability
The raw data supporting the findings of this study are permanently deposited at the Paul Scherrer Institute repository and can be accessed from http://doi.psi.ch/detail/10.16907/409237cf-63de-43ca-b525-a68025a93d63. The derived data, including the reconstructed ptychographic projections and the reconstructed tomogram, are available at the same repository and can be accessed from http://doi.psi.ch/detail/10.16907/a961ab58-de00-40b4-ad6c-bc096776c476. Further details and any other data related to this study are available from the corresponding author upon reasonable request.
Code availability
The code for data reconstruction is available for download directly from cSAXS beamline pages https://www.psi.ch/en/sls/csaxs/software. The package with input/output functions can be downloaded from ‘cSAXS beamline software packages: Base package’. The package for ptychographic reconstruction can be downloaded from ‘cSAXS beamline software packages: PtychoShelves’. The package for tomographic reconstruction can be downloaded from ‘cSAXS beamline software packages: Tomography package’.
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Acknowledgements
This study was financially supported by the Swiss National Science Foundation (Grant Nos. 163220, 188647, 168223 and 190467) and the National Center of Competence in Research Bio-Inspired Materials (Grant No. 51NF40-182881). This project also received funding from the European Union’s Horizon 2020 research and innovation programme (Marie Sklodowska-Curie Grant Agreement No. 706329/cOMPoSe to I.G.). This work was also funded by the European Soft Matter Infrastructure (Grant Agreement No. 731019/EUSMI to J.L. and D.K.) and made use of the Cornell Center for Materials Research Shared Facilities supported by the National Science Foundation’s Materials Research Science and Engineering Centers programme (Grant No. DMR-1719875). U.B.W. thanks the National Science Foundation for financial support (Grant No. DMR-2307013). D.K. acknowledges funding from the Swiss National Science Foundation (Grant No. 200021_175905). J.L. and S.F. acknowledge support from the Japan Society for the Promotion of Science (KAKENHI Grant Nos. 21K04816, 24H00039 and 24H02235), Cooperative Research Projects of the Center for Science and Innovation in Spintronics at Tohoku University and the Graduate Program for Spintronics at Tohoku University. C.D. acknowledges support from the Max Planck Society’s Lise Meitner Excellence Program and funding from the European Research Council (Starting Grant No. 3DNANOQUANT 101116043). U.S. acknowledges funding from the Adolphe Merkle Foundation and the European Research Council (Advanced Grant No. PrISMoID 833895). We further acknowledge the Paul Scherrer Institut, Villigen, Switzerland, for providing synchrotron radiation beamtime at beamline X12SA (cSAXS) of the Swiss Light Source.
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J.L., D.K., A.D. and I.G. conceived the study and supervised the research, with assistance from B.D.W., U.S., S.F. and H.O. The terpolymer was synthesized by T.Y. and H.S. under the supervision of U.B.W. The solvent annealing set-up was designed and built by I.G. and B.D.W. The solvent annealing protocol was developed by K.G. under the supervision of I.G. The polymer template and the gold replica were prepared by S.N.A. The micropillar sample was prepared by M.M. under the supervision of J.L., S.F. and H.O. The PXCT experiment was performed by D.K., M.H., C.D., A.D. and J.L. The ptychographic and tomographic reconstruction was performed by A.D. and D.K. Image processing, domain segmentation and topological charge analysis were performed by D.K. with assistance from K.D., J.L. and I.G. Local structure analysis and structure identification were performed by K.D. The strain analysis was performed by D.K. The manuscript was written by D.K. and J.L. with contributions from all authors, including feedback on results, analysis and interpretation.
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Extended data
Extended Data Fig. 1 Rendering of 2x2x2 unit cell volumes within individual grains.
(a) A slice through the volume (parallel to the substrate) indicating the grain boundary with triangular marks. (b) Slice through the yellow grain perpendicular to the sample surface and its Fourier transform. (c) 3D rendering of the small volume marked with a rectangle in b. (d) Slice through the green grain perpendicular to the sample surface and its Fourier transform. (e) 3D rendering of the small volume marked with a rectangle in d. (f) Slice through the blue grain perpendicular to the sample surface and its Fourier transform. (g) 3D rendering of the small volume marked with a rectangle in f.
Extended Data Fig. 2 Changes to the stripe patterns that result from rotation of the model volume.
(a) 3D rendering and slices through model diamond structure showing the different patterns. (b) 3D rendering and slices through the model rotated by 15° around the x-axis showing the dependence of the patterns on rotation. (c) 3D rendering and slices through the model rotated by 15° around the y-axis showing the dependence of the patterns on rotation. (d) 3D rendering and slices through the model rotated by 15° around the z-axis showing the dependence of the patterns on rotation.
Extended Data Fig. 3 Sample preparation by focused ion beam (FIB).
(a) Scanning electron microscopy (SEM) image showing the sample after FIB milling of its surrounding material. (b) SEM image of the micropillar sample mounted on the OMNY pin. Scale bars are 10 µm.
Extended Data Fig. 4 Principal scheme of the ptychographic experiment.
X-rays with a high degree of coherence are focused by a Fresnel zone plate (FZP) in combination with a central stop (CS) and an order sorting aperture (OSA) to define a coherent illumination onto the sample. X-rays propagate through the sample and are recorded by a 2D detector in the far field. The sample is scanned along x and y for ptychographic imaging, and then rotated in theta for tomographic acquisitions.
Extended Data Fig. 5 Evaluation of the resolution.
(a) 2D resolution estimate of a single ptychographic projection. We show the Fourier ring correlation (FRC) between two ptychographic projections acquired at the same angle as a function of the spatial frequency. The point at which the FRC intersects the threshold curve calculated according to the 1-bit criterion is used as an estimation for the resolution in 2D corresponding to each of the images. (b) 3D resolution estimate of the tomogram. We show the Fourier shell correlation (FSC) between two sub-tomograms, each computed with half of the available projections, as a function of the spatial frequency. The point at which the FSC intersects a calculated threshold for the ½-bit criterion is an estimation of the 3D resolution of the full tomogram, using all the available projections. 2D slices through the tomogram showing (c) XZ, (e) YZ, and (g) XY planes. (d, f, h) Magnified regions are indicated by red boxes in (c, e, g), respectively. (i–k) Line profiles associated with the red dotted lines are shown in (d, f, h), respectively. The numbers indicate the width of the profile line across sharp features of the image using the 10%–90% intensity criterion.
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Karpov, D., Djeghdi, K., Holler, M. et al. High-resolution three-dimensional imaging of topological textures in nanoscale single-diamond networks. Nat. Nanotechnol. 19, 1499–1506 (2024). https://doi.org/10.1038/s41565-024-01735-w
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DOI: https://doi.org/10.1038/s41565-024-01735-w