Soft, malleable double diamond twin

Abstract

A twin boundary (TB) is a common low energy planar defect in crystals including those with the atomic diamond structure (C, Si, Ge, etc.). We study twins in a self-assembled soft matter block copolymer (BCP) supramolecular crystal having the double diamond (DD) structure, consisting of two translationally shifted, interpenetrating diamond networks of the minority polydimethyl siloxane block embedded in a polystyrene block matrix. The coherent, low energy, mirror-symmetric double tubular network twin has one minority block network with its nodes offset from the (222) TB plane, while nodes of the second network lie in the plane of the boundary. The offset network, although at a scale about a factor of 10³ larger, has precisely the same geometry and symmetry as a (111) twin in atomic single diamond where the tetrahedral units spanning the TB retain nearly the same strut (bond) lengths and strut (bond) angles as in the normal unit cell. In DD, the second network undergoes a dramatic restructuring-the tetrahedral nodes transform into two new types of mirror-symmetric nodes (pentahedral and trihedral) which alternate and link to form a hexagonal mesh in the plane of the TB. The collective reorganization of the supramolecular packing highlights the hierarchical structure of ordered BCP phases and emphasizes the remarkable malleability of soft matter.

Keywords: 3D reconstruction; mesoatom; self assembly; soft matter; twin.

Fig. 1.

Identification of TBs in PS-PDMS BCP. (A) A low magnification SEM image showing four adjacent grains with distinct 2D patterns separated by sharp parallel boundaries. (B) Perspective view of the SVSEM 3D data in the laboratory coordinate frame with the boundary plane viewed at an oblique angle and (Right) viewed along the [111] normal. The experimental 2D SEM images are collected parallel to the X-Y plane, and the FIB slicing direction is along Z. (C) SVSEM reconstruction of the interpenetrating PDMS networks spanning the TB viewed along the $\left[11\overline{2}\right]$. The highlighted partial reconstruction is embedded within surrounding translucent networks with the TB plane indicated by the orange line. (D) (Top) ${\left(1\overline{1}0\right)}^{*}$ section of 3D FFT pattern of TB volume SVSEM reconstruction (shown in SI Appendix, Fig. S3C) compared with (Bottom) corresponding pattern for an ideal cubic DD TB. Orange line indicates ${\left[11\overline{2}\right]}^{*}$ reciprocal vector. From the positions of the Bragg peaks, we find that the respective unit cell of each grain (SI Appendix, Table S1) is mildly distorted from cubic, likely due to shrinkage stresses from solvent evaporation during solution casting, as often noted in solution-cast BCP materials.

Fig. 2.

Visualization of the (222) DD TB boundary nodes and symmetry analysis. (A) Visualization of reconstructed PDMS networks (blue and red tetrahedral networks) and the front most TB plane region with aqua and light blue nodes/mesoatoms. The symmetry features of the f = 3 (light blue) and f = 5 (aqua) boundary nodes are clearly identified. (B) Symmetry elements associated with the (222) DD TB plane (central 2D slice from the SVSEM reconstruction). The (222) boundary plane group symmetry is p3m1 with the symmetry elements labeled. The f = 3 nodes are in Wyckoff site 1a, the f = 5 are in 1b, and the red network struts pass through site 1c. (C) Two perspective views of the blue TB network showing nodes with functionality of 3 and 5. An enlarged view of a 6-(5, 3)³ loop in the boundary plane showing the alternating node types. (D) Heat maps of the IMDS normal distribution function averaged over all f = 3 (Top) and f = 5 (Bottom) nodes in the TB. These distributions show both symmetry about the equator ($\theta , \phi$ chart on left) as well as 120-degree rotational symmetry about [111] (stereographic projection, right), consistent with $\overline{6}m2$ point group symmetry.

Fig. 3.

Geometry and topology of the DD TB. (A) SVSEM reconstructions of the three types of boundary loops. In the red network, there are no boundary nodes, while in the blue network, two new types of boundary nodes (f = 3 and f = 5) are formed. (B) Skeletal graph of the red network displaying the set of parallel struts that cross normal to the boundary. The skeletal graph of the blue network shows the network nodes coincide with the plane of the boundary. Consequently, the network reorganizes into a 2D periodic array of 6-(5, 3)³ loops. The catenated nature of the blue and red networks is best understood from viewing Figs. 1C and 2A.

Fig. 4.

Malleability for accommodation of the TB. (A) IMDS structures of experimental SVSEM reconstruction showing positions of various structural features in the respective red and blue network layers along the perpendicular to the (222) TB. (B) IMDS based on SVSEM data showing new boundary nodes and boundary struts in the blue and red networks. In the blue network, there are two types of struts: 5–3 struts along <110> directions and 5–4 struts along the [111] direction. There are two types of boundary internode angles for the f = 5 nodes: Angle 4-5-3 = 90°, Angle 3-5-3=120°; and one type for f = 3 nodes: Angle 5-3-5=120°. In the red network, there is no change in node connectivity, the boundary strut is perpendicular to the TB oriented along the [111] direction. (C) Internode angle distribution of nodes in different groups versus the distance between nodes and TB plane (for each data point, at least 78 angles are measured). (D) Node volume distribution in different groups versus the distance between nodes and TB plane (for each data point, at least six nodes are measured). (E) Node surface-to-volume ratio distribution in different groups versus the distance between nodes and TB plane (for each data point, at least six nodes are measured). Distribution of strut lengths in and normal to the TB: (F) blue network boundary f = 5 node to off boundary f = 4 node length distribution measured from 52 struts; (G) blue network f = 5 node to f = 3 node boundary strut length distribution measured from 86 struts; (H) red network strut length distribution of f = 4 node to f = 4 node struts which are perpendicular to TB plane versus the distance between the strut centroid and the TB plane (for each data point, at least seven struts are measured). Error bars represent a SD about the averaged value.