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Seeing mesoatomic distortions in soft-matter crystals of a double-gyroid block copolymer


Supramolecular soft crystals are periodic structures that are formed by the hierarchical assembly of complex constituents, and occur in a broad variety of ‘soft-matter’ systems1. Such soft crystals exhibit many of the basic features (such as three-dimensional lattices and space groups) and properties (such as band structure and wave propagation) of their ‘hard-matter’ atomic solid counterparts, owing to the generic symmetry-based principles that underlie both2,3. ‘Mesoatomic’ building blocks of soft-matter crystals consist of groups of molecules, whose sub-unit-cell configurations couple strongly to supra-unit-scale symmetry. As yet, high-fidelity experimental techniques for characterizing the detailed local structure of soft matter and, in particular, for quantifying the effects of multiscale reconfigurability are quite limited. Here, by applying slice-and-view microscopy to reconstruct the micrometre-scale domain morphology of a solution-cast block copolymer double gyroid over large specimen volumes, we unambiguously characterize its supra-unit and sub-unit cell morphology. Our multiscale analysis reveals a qualitative and underappreciated distinction between this double-gyroid soft crystal and hard crystals in terms of their structural relaxations in response to forces—namely a non-affine mode of sub-unit-cell symmetry breaking that is coherently maintained over large multicell dimensions. Subject to inevitable stresses during crystal growth, the relatively soft strut lengths and diameters of the double-gyroid network can easily accommodate deformation, while the angular geometry is stiff, maintaining local correlations even under strong symmetry-breaking distortions. These features contrast sharply with the rigid lengths and bendable angles of hard crystals.

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Fig. 1: Supra-unit cell structure of the PS-PDMS double-gyroid tubular network phase.
Fig. 2: Sub-unit cell IMDS curvature and distance metrics.
Fig. 3: Sub-unit cell length and angular metrics.
Fig. 4: Models of sub-unit cell morphology.

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SVSEM and SCF modelling data are available at

Code availability

Supporting software codes are available at


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Primary support for this research was provided through the US Department of Energy (DOE), Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under award DE-SC0014599 to G.M.G. and E.L.T. Use of the Advanced Photon Source at the Argonne National Laboratory was supported by the US DOE, Office of Science, and Office of Basic Energy Sciences. A grant from the National Science Foundation to E.L.T. under award DMR 1742864 supported the development of SVSEM techniques. A grant from the Ministry of Science and Technology supported the R.-M.H. group. SCF calculations were performed on the UMass Cluster at the Massachusetts Green High Performance Computing Center. We thank B. van Leer, T. Lacon and T. Santisteban from Thermo Fisher for insights into the hardware and software underlying SVSEM tomography.

Author information

Authors and Affiliations



The research was designed and supervised by E.L.T. and G.M.G. The BCP was synthesized by A.A., and R.-M.H. was responsible for processing samples. The SVSEM technique was developed by H.G., M. Z. and X.F., and carried out by X.F. and H.G. K.Y. provided modifications to the SVSEM software. Numerical algorithms for morphology analysis were developed by C.J.B. with I.P., and applied by C.J.B., X.F. and A.R. SCF computations were carried out and analysed by A.R. The manuscript was written by E.L.T, G.M.G., X.F., C.J.B. and A.R.

Corresponding authors

Correspondence to Gregory M. Grason or Edwin L. Thomas.

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Extended data figures and tables

Extended Data Fig. 1 Variation in the unit-cell parameters of triclinic unit cells within one grain.

Unit-cell parameters at five different places (1–5) within one grain are measured in real space. The result indicates that the structure is coherent but non-cubic, with unit-cell parameters exhibiting only small deviations throughout the many-cubic micrometre grain.

Extended Data Fig. 2 Strain eigenvector mapping from a cDG lattice to a vtDG lattice within the slicing coordinate frame of reference.

Directions and magnitudes of deviations from cubic symmetry in different grains of the sample are not correlated with the ion-milling (slicing) direction (Z).

Extended Data Fig. 3 Mean (H) and Gaussian (K) distribution of IMDS curvature in theoretical models.

a, b, Distributions are shown for a constant mean curvature (CMC) surface (a) and for a constant matrix thickness (CMT) surface (b). cf, Distributions obtained from SCF theoretical calculations of cDG as a function of segregation strength: χN = 12.5 (c), χN = 15 (d), χN = 25 (e) and χN = 35 (f). As in the main text, D is the cubic unit cell repeat length.

Extended Data Fig. 4

Histogram showing the internode angles of experimental vtDG (from SVSEM) and non-affine triclinic SCF models.

Extended Data Fig. 5 SAXS pattern from a region of the bulk polygranular PS-PDMS sample.

The structure can be nominally associated with a double-gyroid morphology, with an average cubic lattice parameter of D = 130 nm. Diffraction from the {110}tDG and {200}tDG families, which are forbidden for the cubic \(Ia\bar{3}d\) space group, are observed, indicating the non-affine deformation of the cubic double-gyroid lattice.

Extended Data Fig. 6

The workflow for collection and analysis of SVSEM tomography data. See Methods for more details.

Extended Data Fig. 7 Acquisition and processing of SVSEM images.

a, Illustration of the SVSEM reconstruction method. In step 1, low-energy incident electrons (1 KeV) are used to image the near-surface region of a bulk sample. In step 2, a Ga+ beam is used to slice a roughly 3-nm-thick section from the sample surface. In step 3, electrons are again used to image the ion-beam-milled sample surface. The process is repeated. With a large enough number of images (more than 200), the 3D morphology can be constructed via alignment of the stack of slices. b, Different sample stage positions are used for undistorted imaging during slice-and-view. i, For ion milling during slicing, the sample observing surface is parallel to the ion beam. ii, For electron imaging, the sample observing surface is perpendicular to the electron beam.

Extended Data Fig. 8 Secondary-electron images acquired with different electron-accelerating voltages.

Corresponding raw greyscale pixel-intensity distributions are presented blow the electron images. With a lower accelerating voltage, there is a clear binary separation of the pixels into a dark peak (left) and a bright peak (right). Each image is from a freshly sliced region.

Extended Data Fig. 9 Alignment fiducials and monitoring of slice thickness during experiments.

a, An X-shaped fiducial (within the red square) in ion-beam view is used for the registration of FIB slicing. b, An X-shaped fiducial (within the yellow square) in electron-beam view is used for registration of electron imaging. The round cross-section of the perpendicularly drilled hole (within the orange square) is used for the fine registration of secondary-electron images. c, A secondary-electron image used for data acquisition, showing an ion-milled hole (within the orange square) used for fine registration. d, Monitoring of slice thickness: we measured the distance between the milling surface of the nth slice and the milling surface of the first slice (total slice thickness d) from FIB images, and then plotted d versus (n − 1). This reveals a linear relationship with a slope of 2.96 ± 0.01, which is the averaged slice thickness (in nm).

Extended Data Fig. 10 Monitoring of potential SEM image rotation during FIB-SEM image collection.

a, SEM raw data image of the region of interest, with two holes drilled normal to the slice surface by the FIB. b, c, Side-view snapshots of 3D reconstructed holes 1 and 2 (the corresponding rotational videos are in Supporting Video 3). The image stack (80 slices) was aligned using hole 1. The reconstruction of hole 2 is still symmetric, indicating no image rotation.

Extended Data Fig. 11 Important Fourier components of the experimental double-gyroid structure.

The overall intensity of each diffraction family was normalized by the strongest {211}tDG family from the 3D FFT of the region containing approximately 160 unit cells (the same region as in Fig. 4f, which shows the intensity of each individual peak). The overall normalized intensity data are plotted against their associated cubic q value for those planes. For our reconstructions, we use a Bragg filter that selects peaks above an intensity threshold of 10−7 (Imin) for associated q values smaller than 0.2 nm−1.

Supplementary information

Supplementary Video 1

Comparison video of slicing along c direction of the 2 × 2 × 2 unit cell volume box (Fig. 1b) and slicing of corresponding SCF tDG.

Supplementary Video 2

Rotational view of a rendering of 3D FFT analysis from a region containing approximately 160 unit cells (same region as Fig. 1c) before Bragg filtering.

Supplementary Video 3

Rotational view of 3D reconstructions of hole 1 and hole 2, which are presented in Extended Data Fig. 10.

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Feng, X., Burke, C.J., Zhuo, M. et al. Seeing mesoatomic distortions in soft-matter crystals of a double-gyroid block copolymer. Nature 575, 175–179 (2019).

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