Gradient Transformation of the Double Gyroid to the Double Diamond in Soft Matter

Abstract

Transitions between gyroid and diamond intercatenated double network phases occur in many types of soft matter, but to date, the structural pathway and the crystallographic relationships remain unclear. Slice and view scanning electron microscopy tomography of a diblock copolymer affords monitoring of the evolving shape of the intermaterial dividing surface, allowing structural characterization of both the majority and minority domains. Two trihedral malleable mesoatoms combine to form a single tetrahedral mesoatom in a volume additive manner while preserving network topology, as the types of loops, the number of mesoatoms in a loop, minority domain strut lengths, and directions that connect a given mesoatom to its neighbors evolve across a 150 nm wide transition zone (TZ). The [111]_DD direction is coincident with the [110]_DG direction so that the (111)_DD and (110)_DG planes define the boundaries of the TZ. Selection of the particular crystal orientations and direction and width of the transition zone is to minimize the cost of morphing the mesoatoms from one structure to the other, by maximizing like-block continuity and minimizing the variation of the surface curvature and thickness of the domains across the TZ. Such coherent continuity of the independent, intercatenated networks across the transition zone is critical for applications such as graded mechanical trusses where the pair of different networks are joined to provide different mechanical properties for adjacent grains or could serve as a nanoscale anode/cathode allowing super charging and discharging provided the networks are continuous and rigorously separate.

Introduction

Studying how the atoms maneuver during a phase transformation can be beneficial in relating microstructure to property performance.¹⁻⁵ Following the details of phase transformations in soft matter is, however, more challenging than in atomic crystals.⁶ Indeed, for bicontinuous network structures, such as the double diamond (DD)^7,8 and double gyroid (DG),^9,10 it is not even clear how to identify and monitor the basic structural units that are involved in the transition, since there are no discrete motifs. In lipid–water systems, the amphiphilic lipid molecules form a continuous bilayer that surrounds a pair of 3D continuous water network channels, while in block copolymers (BCPs), the majority block forms a continuous matrix surrounding the two continuous minority block domains. The intermaterial dividing surface (IMDS) constitutes the continuous interface between the majority–minority block–block domains.¹¹ Each tubular network in the DG is a 10-3¹⁰ net and in the DD, a 6-4⁶ net.¹² Phase transformations¹³⁻²⁸ between network phases are commonly observed in surfactant–water systems,^13,18,19,23 thermotropic liquid crystals,²⁴ block copolymers,^27,28 and BCP–homopolymer blends.^25,26 Researchers working on lipid membrane structures initially speculated the DD ⇔ DG transformation might follow the Bonnet transformation which takes one topologically equivalent triply periodic minimal surface (TPMS) (i.e., Schwarz’s D) to the other (i.e., Schoen’s G).¹³ The ratio of the lattice parameters of the two cubic phases in experiments is sometimes ∼1.6 which is nearly that (1.576) predicted for the Bonnet transformation.^15,22 However, except for the starting and ending TPMS, all other members of the D to G Bonnet surface family exhibit physically unrealistic self-intersections,^14,19 which require fragmentation of the initial continuous lipid membrane structure followed by fusion into the other continuous structure.¹⁸

Wells,¹² from his considerations of periodic graphs, first pointed out that the diamond net can be converted into the gyroid net (and vice versa) via replacement of each 4 node with a pair of connected 3 nodes. Sadoc and Charvolin¹⁴ speculated that, in inverse bicontinuous surfactant–water systems, the diamond network can be transformed to the gyroid network by changing a tetrafunctional node into a pair of trifunctional nodes. Subsequently, several continuous, uniform deformation pathways between DD and DG were proposed.^16,17,21 One pathway suggested by Fogden and Hyde¹⁷ involves continuously deforming the cubic diamond net into a tetragonal unit cell via an extension along the [100]_DD direction, causing splitting of each four-strut node of the diamond network into two connected three-strut nodes. X-ray signatures of the initial and final phase orientations are consistent with this proposed DD → DG pathway (or its reverse, DG → DD). For example, Oka²³ used X-ray diffraction to find the initial and final crystal orientations for a lipid–water system consisting of one DD grain and one DD grain and adopted the model that uniformly separates tetrahedral nodes along [100]_DD into trihedral nodes and in addition required one of the ⟨111⟩_DD water channels to be aligned with one ⟨110⟩_DG water channel. Chang et al.²⁸ investigated the same polystyrene-b-polydimethylsiloxane (PS-PDMS) BCP as in our present study and, by rapid evaporation of a selective solvent followed by thermal annealing, observed using SAXS and TEM tomography a series of transformations from the double P (Plumber’s Nightmare) network phase to the DD to the DG. They also invoked the splitting of the tetrahedral node into 2 trihedral nodes and proposed an epitaxial relationship between DD and DG but only specified that the [211]_DD direction was parallel to the [110]_DG. Thus, while the splitting of a tetrahedral node into a pair of trihedral nodes has been given as the mechanism for the transformation by many authors, this is a local view without information about how the growing crystal reorients or any specification of possible epitaxial relations across the assumed sharp DD–DG phase boundary.

Results and Discussion

The DG to DD Transformation

Using slice and view scanning electron microscopy (SVSEM), we reconstructed the DG–DD phase boundary in three separate microspheres of a polystyrene-b-polydimethylsiloxane (PS-PDMS) diblock copolymer and found the same crystallographic relationship between adjacent grains of DG and DD. Our sample preparation, microscopic techniques,²⁹ and the indirect solvent annealing procedure used to drive the phase transition are detailed in the Methods section. Under the indirect solvent annealing processing conditions used, the phase transition direction proceeds from the DD into the DG structure. The 3D SVSEM reconstruction in Figure 1A includes approximately 150 DD and 30 DG unit cells and reveals an approximately 150 nm wide transition zone (TZ) over which the structure changes continuously from DG to DD. In the grayscale SEM image (Figure S1), the bright regions correspond to the PDMS domains and the dark regions, to the PS domains. Segmentation and registration of ∼100 such images after serial imaging and focused ion beam slicing, allows 3D reconstruction over a large volume. By setting the PS matrix as transparent, one directly views the IMDS that surrounds the two pairs (red/blue) of 3D continuous intercatenated PDMS networks which are continuous across the TZ. In order to follow the details of the transformation, we use the mesoatom concept to define discrete building blocks of each tubular network phase.³⁰ The trihedral mesoatom and tetrahedral mesoatom are the fundamental units of the respective cubic DG and DD phases with each mesoatom having an inner portion composed of the minority block surrounding the inner terminal surface and an outer portion composed of the majority block surrounding the IMDS and extending to the outer terminal surface (Figure S2). The DG and DD mesoatoms occupy Wyckoff sites 16b and 2a within the respective bcc (space group Ia3d) and primitive cubic (space group Pn3m) unit cells.³⁰ In our BCPs, a mesoatom typically contains thousands of BCP molecules and millions of atoms.

Figure 1.

Transformation zone of double gyroid to double diamond. (A) Reconstructed volume of the DD, TZ, and DG regions. (B) Higher magnification reconstruction of the white region outlined in (A). The IMDS that encloses the intercatenated minority PDMS domains is rendered red or blue with the PS matrix rendered transparent. The various types of loops within the TZ are labeled with numbers. Note that we use the m–f^y notation to indicate the smallest closed loop in the network graph consisting of m nodes with each node having f connected neighbors and the exponent y, indicating the number of nodes in the smallest fundamental loop. The 10 mesoatom DG loops change into loops consisting of 9, then 8, then 7, and finally become the 6 mesoatom loops of the DD phase. Blue circles with interior symbols indicate the approximate orientation of the trihedral DG mesoatom struts. Symbols: uniform: struts are all in-plane; black dot: one strut directed upward out of the plane; black X: one strut directed downward out of the plane. Yellow circles indicate the orientation of the DD struts: a yellow circle with a black dot symbol indicates one strut directed normal upward, and a yellow circle with a black X symbol indicates the strut is directed normal and downward. The pairs of yellow lines indicate the boundary planes of the TZ which extends from a (111)_DD plane to a (110)_DG plane. (C) Intercatenated loops in the DD, TZ, and DG. This figure corresponds to red slab 3 of the series of slabs shown in Figure S8. The light blue numbers in the center indicate the number of mesoatoms in a loop, and the numbers 3 and 4 indicate the functionality of the mesoatom. The set of inclined intercatenated blue loops are extracted and viewed at the right side to show the type and number of mesoatoms in each loop.

The number and type of mesoatoms in the fundamental loops comprising the two networks vary across the TZ as shown in Figure 1C where the 10 node DG loops at the right side change to loops of 9, 8, and 7 mesoatoms and finally to the 6 node DD loops at the left side for both the red and blue networks. The outer boundary planes of the TZ are shown as yellow lines in Figure 1B and correspond approximately to (111)_DD and (110)_DG, beyond which the structure resumes the normal DD or DG crystal symmetries. Important symmetry directions and the coordinate axis directions for the DD and DG unit cells are labeled for an isolated slab of the structure spanning from the DD grain across the TZ to the DG grain in Figure 2A. The set of connected DD mesoatoms along [011]_DD remains parallel to the set of connected DG mesoatoms along [001]_DG (Figure 2B). Viewing successive pairs of these adjacent, parallel mesoatom spirals across the TZ shows how the DG mesoatoms cooperatively vary their distances and orientations along their 4₁ and 4₃ screw axes and merge to become DD mesoatoms, forming the neutral 2₁ DD screw axes approaching the DD side of the TZ. Viewing a succession of 5 loops across the TZ also shows the detailed structural variations of the mesoatom nodes and struts for the red network in Figure 2C using 6 colors to define the apolar ⟨110⟩_DG strut directions (e.g., [110] = [110]). The 3 pairs of green, purple, and red DG struts in the 10-3 DG loop evolve to become the set of 6 struts comprising the 6-4 DD loop, while the pairs of blue and yellow DG struts shorten and vanish across the TZ as the DG mesoatoms at their ends merge together to form a DD mesoatom. Other informative views of the transformation across the TZ are given in the Supporting Information (especially Figures S4–S8). Since the transformation is occurring by advancement of the DD grain into the DG grain, the trihedral mesoatoms that are closest to the DD grain evolve first (see Figure S4) forming the intermediate node loops composed of 3 and 4 strut mesoatoms. The network topology remains unchanged as the intercatenated red/blue chiral DG networks connect smoothly to their respective intercatenated red/blue achiral DD networks.

Figure 2.

Cooperative mesoatom interactions occurring during the DG to DD transformation. (A) (top) Single network slab spanning from the DD crystal across the TZ to the DG crystal. The location and orientation of various rotational and screw symmetry axes are indicated. One 6-4⁶ DD loop and one 10-3¹⁰ DG loop are highlighted in dark red color along with a small central portion of the corresponding intercatenating dark blue network. (bottom) The network slab has been rotated by 90° to show the location of symmetry axes. The TZ is noncrystallographic but does convey symmetries between the phases as well as continuity of the pairs of networks. (B) A view of mesoatoms along the special strut directions ([111]_DD and [110]_DG). At the rightmost and leftmost sides are level set models showing the characteristic screw symmetries of DD (2₁) and DG (4₁ and 4₃) with 3 experimental reconstructions of corresponding regions within the TZ revealing the evolution from DG to DD. (C) Series of extracted loops showing how the mesoatoms in the loops vary across the TZ from 10-3¹⁰ through 9-4,3⁸; 8-4⁴3⁴; 7-4⁵3²; to 6-4⁶. The DG loop progressively transforms to a DD loop starting with the mesoatoms nearest the DD grain. The strut colors indicate the particular ⟨110⟩_DG direction. (D) Simple schematic representation of the volume preserving mesoatom reaction whereby a pair of neighboring trihedral DG mesoatoms merge to form a single tetrahedral DD mesoatom. Tomographic reconstruction of the evolution across the TZ of the merging of a pair of DG mesoatoms (highlighted in red).

Monitoring the transformation by following the mesoatoms provides insight into how the network chirality synchronizes across the TZ. As no minority network is in direct contact with any of the others, primary communication of the morphological transition is via the surrounding outer majority component PS brushes that contact across the outer nonconvex terminal surface of the mesoatoms (see Figures S2 and S5). Considering the reverse DD → DG transformation, the emerging network chirality starting from an initial achiral diamond network would be determined by the leading portion of the transformation front, as once a pair of DG mesoatoms are created, complementary matching of curvatures across monkey saddles of the outer terminal surfaces of neighboring mesoatoms in concert with their neighboring mesoatoms necessitates alternating, opposite chirality nonlinked neighbors as the transformation front moves forward (Figure S6). The transformation consists of a variety of mesoatom adjustments within the TZ. Overall, the proposed mechanism where pairs of trihedral coordinated nodes in a 10 node gyroid loop merge to form tetrahedral coordinated nodes in a 6 node diamond loop occurs but in a much more complicated fashion than earlier supposed.

Since there are two DD mesoatoms per DD unit cell and 16 DG mesoatoms per DG unit cell (Figure S2), based on the measured lattice parameters (Figure S3), the DD mesoatom has 2 times the volume of a DG mesoatom (i.e., one unit cell of 16 DG mesoatoms transforms to 4 unit cells containing 8 DD mesoatoms). Thus, the experimental lattice parameter ratio, a_DG/a_DD = 1.6 ≈ ³√4 reflects the mesoatom–mesoatom volume preservation reaction rather than indicating some connection of the phase transition to the Bonnet transformation. Software extraction of the intermediate mesoatoms indeed demonstrates that the volume enclosed by a pair of merging mesoatoms is approximately constant over the TZ. Thus, the DG to DD phase transformation can be considered as a type of “mesoatom reaction” as schematically shown in Figure 2D.

The real space 3D tomographic visualization also allows the determination of the crystallography of the phase transition to be discerned by determining the various strut directions in the adjacent DD and DG grains. This can be done by analyzing the real space 3D reconstruction and the 3D Fast Fourier Transform (FFT) diffraction patterns of the selected regions. Importantly, real space data afford translational registration information beyond just orientational relations available from diffraction. Analysis of the skeletal graphs shown in Figure 3A reveals that the [111]_DD is coincident with the [110]_DG and that [011]_DD is parallel to the [001]_DG. This first relation aligns one [111]_DD strut direction with one [110]_DG strut direction, and the second relation is relevant to the efficient conversion of left and right handed 4-fold DG screw axes into the neutral 2₁ screw axes of the DD (Figure 2B). The length distribution of a large number of struts is shown in Figure 3B (∼200 struts from the DD region, ∼300 from the DG region, and ∼400 from the TZ). The strut length distribution in the TZ includes the shortening (and disappearing struts) and the lengthening and reorienting struts. In order to appreciate the orientational relationships of the various strut directions in the respective DG and DD crystals and especially in the TZ, we employed a stereographic projection (Figure 3C). This construction sets the relationship of the two grains by superposing a unit cell representing each grain at the center of a sphere and extending directions and planes until they intersect the surface of the sphere, then projecting these points and curves onto the equatorial plane (the stereographic projection). Directions intersect the sphere surface as points and planes intersect as great circles. The experimental distribution of strut directions is shown in Figure 3D, where the location of a point in the stereographic projection provides the strut orientation, and the color of the point indicates the strut length. The DD and DG unit cells are placed such that [111]_DD and [110]_DG directions are coincident, and we find that a 35.27° rotation about the [110]_DG axis brings [011]_DD parallel to [001]_DG. Thus, the five degrees of freedom³¹ of the phase–phase boundary can be specified as 35.27°, [110], (111)_DD. Oka’s²³ orientational relationships between DD and DG for the lipid–water system are entirely consistent with this assignment, suggesting that a wide class of bicontinuous network structures may exhibit very similar structural behavior during phase transitions.

Figure 3.

Symmetries of the phase–phase boundary. (A) Skeletal graphs of the DD, TZ, and DG regions with the length of the strut indicated by the color. The longer DD struts are blue, and the shorter DG struts are green. The thin black lines indicate the approximate borders of the TZ and also locate facets along the transformation front. The thicker vertical black line shows the coincident [111]_DD and [110]_DG strut directions. The [001]_DG is parallel to the [011]_DD, and both are orthogonal to the strut coincidence direction. (B) Histogram distribution of the 900 strut lengths. (C) Stereographic projection of the important directions in the DD and DG crystals, with the coincident [111]_DD and [110]_DG located in the center. (D) Stereographic projection of the experimental DD, DG, and TZ strut directions shown as colored dots (proportional to strut length) for the 6 ⟨110⟩_DG directions (blue lens symbol) compared to the closest ⟨111⟩_DD direction (gold triangle symbol).

Monitoring the evolution of the directions and lengths of the struts connecting neighboring mesoatoms across the TZ explicitly demonstrates that each four-strut node of the diamond network (1 tetrahedral mesoatom) is formed by the merger of two three-strut nodes (2 trihedral mesoatoms). This process occurs via local mesoatom interactions, resulting in a gradient structure across the TZ that does not possess translational symmetry. The proposed models of the transformation that assume progressive, uniform deformation of crystallographic cells are incorrect.^16,17,21 Strict homoepitaxy between the DD and DG structures, wherein the lattice of the newly formed phase is in precise registry with the substrate lattice across a sharp phase–phase boundary, does not occur. Instead, we find a type of “gradient epitaxy”, with an ordered but nonperiodic gradient transition zone having a width of about 150 nm that seamlessly connects the DD and DG networks. Crystallographic information is clearly transmitted across this zone, as certain rotational symmetries are aligned (see Figure 2A), and importantly, the network topology is conserved as the two opposite chirality 10 node DG networks each smoothly change into a pair of achiral 6 node DD networks. The directions and lengths of the struts progressively evolve as the TZ propagates with mesoatoms reconfiguring from DG to DD. Referring to the 6 apolar directions of the ⟨110⟩_DG family shown with individual colors in Figure 2C, one notes three types of behavior: (i) the family of [110]_DG struts (green) that is coincident with the [111]_DD direction maintains a tight orientational grouping since these struts do not reorient but only grow longer, (ii) three other ⟨110⟩_DG strut families (brown, red, and purple) lengthen and reorient toward the closest ⟨111⟩_DD direction (theoretically, the [101] and [101] directions rotate 10.53° and the [110], rotates 19.47°), while (iii) the remaining two ⟨110⟩_DG strut families (yellow, blue), which have the largest angle (>45°) with respect to any ⟨111⟩_DD direction, shorten and eventually disappear, indicative of merging of pairs of DG mesoatoms into a single DD mesoatom.

In phase transitions, the ability to minimize the nucleation barrier and resultant phase boundary energy are paramount in the selection of the transformation pathway.³¹ The high fidelity 3D SVSEM data unambiguously reveal the crystallographic details of the DG to DD transformation pathway, but why is this particular pathway favored? It appears that the transformation picks a route that minimizes the necessary reconfiguration of the mesoatoms by selection of one ⟨111⟩_DD axis to be coincident with one ⟨110⟩_DG axis as optimal for aligning/transforming the sets of network struts for this flexible block copolymer or for lipid–water systems.²³ The transformation of a material having rigid units or fixed strut lengths leads to other pathways for the DG–DD transition involving noncubic intermediate structures.²⁴ A video (see the Supporting Movie) made by software slicing across the reconstructed TZ region normal to the coincident [111]_DD and [110]_DG directions shows that the PDMS and PS domains evolve smoothly across the TZ with continuous adjustments of the mesoatoms from trihedral to tetrahedral (see Figure S7 for a series of single frames). This matching of domain patterns is a 3D soft matter version of the 2D matching of coincident site lattices which leads to good atomic fit across sharp, planar grain boundaries and correspondingly low grain boundary energy in hard matter crystals.³¹ Indeed, the malleable mesoatoms can adapt to provide excellent PDMS–PDMS and PS–PS matching across the TZ at a relatively small energetic cost. Interestingly, and in accord with hard matter, occasional facets (steps) on the boundary of the TZ are also observed indicating that the transformation is locally occurring in two directions (see parallel zigzag black lines in Figure 3A). The main transformation is along the coincidence direction while the lateral motion of facets occurs along a different ⟨111⟩_DD type direction (one having ∼10° misorientation with ⟨110⟩_DG). The steps appear to be about 1–2 strut lengths in height.

Conclusions

Many directions remain to be explored concerning possible gradient epitaxy transition zones in other order–order phase transformations of soft matter. The selection of one strut direction in the transforming phase to be coincident with that of the growing phase with lattice rotation about this direction in order to minimize the required reorganization of the transforming mesoatoms should prove a useful concept for future theoretical computations of optimal crystallographic relations and minimum energy structural transition pathways between other complex BCP and lipid–water tubular network phases, e.g., the DP to DD transformation.^28,32−34 The facile interconversion of the two types of mesoatoms in a volume additive manner and the seamless continuity of their respective networks across the phase–phase boundary shows the adaptability of malleable trihedral and tetrahedral mesoatoms which suggests the likelihood of finding other periodic networks composed of both types of mesoatoms (e.g., Wells’ periodic mixed 3,4 nets¹²).

Methods

Materials and BCP Microsphere Fabrication

The polystyrene-b-polydimethylsiloxane (PS-PDMS) diblock copolymer was synthesized by sequential anionic polymerization of styrene and hexamethylcyclotrisiloxane.²⁸ The number-average molecular weights of the PS block and the PDMS block are 51 and 35 kg/mol, respectively. The volume fraction of the PDMS block is ∼42%, and polydispersity of the sample is 1.05. Monodisperse microspheres were made from an emulsion of 6 mg/mL polymer–toluene droplets in a continuous phase of DI water containing sodium dodecyl sulfate (SDS, 0.1 wt %). With the help of nitrogen gas, the polymer solution was passed through 5.1 μm diameter pores of a Shirasu Porous Glass membrane, forming near-monodisperse microspheres stabilized by the surfactant. The concentration of toluene in the suspended microspheres decreased very slowly over 3 days due to its limited solubility in the water matrix. Subsequently, the microspheres were dispersed onto a glass slide and coated with a 50 nm protective layer of platinum for slice and view SEM. FIB slicing indicated that approximately 10% of the spheres (20 spheres) were DG and that 90% exhibited regions of both DG and DD.

Indirect Solvent Annealing

The schematic of the apparatus used is shown in Figure S9. Ten milliliters of water solution containing BCP microparticles was transferred into a 20 mL vial containing a stir bar. The vial was then placed into a 100 mL bottle containing 3 mL of chloroform. The containing bottle was then sealed with a cap and parafilm allowing the entrapped chloroform vapor to slowly dissolve into the water and subsequently absorb into the suspended BCP microparticles, lowering the T_g of the PS domains and allowing the phase transformation from DG to DD to proceed. Samples were recovered from this treatment over periods from 1 to 7 days. Then, the 20 mL vial was removed, and the remaining solvent slowly evaporated for another 2 days (vial open, stirring continuously). Then, the indirect solvent annealed microspheres with water and surfactants were placed on a glass substrate and allowed to dry. Finally, the glass slide with the microparticle BCP samples was coated with a 50 nm layer of platinum for slice and view SEM (Figure S10). FIB slicing revealed that 10% of the indirect solvent annealed spheres (20 spheres) exhibited regions of DG and DD and 90% had fully transformed to DD.

Slice-and-View SEM

The acquisition of SVSEM data follows the procedure previously reported.²⁹ A Thermo Fisher Helios NanoLab G4 CX SEM-FIB DualBeam system employing a gallium ion beam (Ga⁺) having an energy of 30 keV with a current of 80 pA was used to mill 3 nm slices from the sample surface. A 1 keV electron beam with a beam current of 48 pA was used to image the sample surface using a Through Lens (TLD) secondary electron (SE) detector. The stronger scattering from the higher atomic number of Si atoms in the PDMS and the resulting additional SE emission are sufficient to provide excellent secondary electron intrinsic contrast between the PS and PDMS domains without staining. Fiducials were used to register the FIB and SE images during the automatic slice and view process. The pixel resolution for each 2D SEM image collected is 3.0 nm/pixel. The FIB slice thicknesses during the image acquisition process were monitored based on FIB images as 3.03 ± 0.1 nm/slice; thus, a specific (3 nm)³ cube for our voxel resolution was used in the reconstructions.

Data Processing and Analysis

ImageJ (https://imagej.nih.gov/ij/) was used to binarize the grayscale image stack data to identify the tubular networks formed by the minority PDMS domains and separate them from the majority matrix via a threshold intensity value chosen to match the volume fractions of the two binary components with the experimentally reported PDMS block volume fraction (∼42%). Reconstructions and morphological analysis were done with Avizo software from Thermo Fisher and Supporting Software as reported.²⁹ Skeletonization allows identification of mesoatom nodes, functionality, strut lengths, and directions. We extracted the skeleton directly from the binarized 3D volume with the skeletonization feature in ImageJ to thin down the IMDS to create a graph having a set of nodes at the mesoatom centers and edges, which represent the length of struts connecting neighboring mesoatoms. The skeletal data were then imported into the Mathematica program AnalyzeSkeleton to obtain the distributions of the strut length and direction.

3D FFT Analysis

Information on unit cell vectors was extracted from the 3D FFT of the SVSEM data set. Before transformation of the real space volume data into reciprocal space, a Hanning window was applied to the raw SEM data in order to reduce artifacts in the FFT associated with sample edge boundary discontinuities. The unit cell vectors are obtained from the indexed 3D FFT patterns in the respective DD and DG grains using approximately the same sample volume, corresponding to ∼40 DD and ∼10 DG unit cells.

3D Reconstruction Visualization

Based on the binarized images, a 3D volume can be reconstructed by using Avizo software. Subvolumes of regions of interest (ROI) can be further cropped out for local analysis. Skeletonization allows identification of mesoatom nodes, functionality, strut lengths, and directions.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsnano.3c11101.

• Supplementary text, grayscale SEM images, schematics of the trihedral and tetrahedral mesoatoms and unit cells, unit cell parameter calculations, schematic of loop transformations, 3D data of matrix PS with PDMS networks transparent, schematic of mesoatom interaction on chirality, a series of software slices showing 2D matching of coincident site lattices in DD and DG, 3D data on intercatenated loops in the DD, TZ, and DG, schematic of indirect solvent annealing procedure, and cross sectional SEM images showing coexistence of DD and DG in multiple microspheres (PDF)

• Related ion beam slicing and electron beam imaging from DG to TZ to DD (MOV)

Author Contributions

W.S. and E.L.T. designed the research; W.S. performed the research; together W.S. and E.L.T. analyzed the data and wrote the paper.

This research was principally supported by NSF DMR under award DMR 2105296 as well as a DOE Basic Energy Sciences award DE-SC0022229.

The authors declare no competing financial interest.