Liquid Organic Frameworks: The Single-Network &quot;plumber&#39;s Nightmare&quot; Bicontinuous Cubic Liquid Crystal

Nightmare ” Bicontinuous Cubic Liquid Crystal

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Liquid Organic Frameworks: The Single-Network “Plumber’s

Silvio Poppe, Xiaohong Cheng, Changlong Chen, Xiangbing Zeng, Rui-bin Zhang, Feng Liu,*

Goran Ungar,* and Carsten Tschierske*

Cite This: J. Am. Chem. Soc. 2020, 142, 3296 −3300

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sı Supporting Information

ABSTRACT: Novel bolapolyphiles, built of a p-terphenyl or bistolane core with polar glycerol end-groups and two laterally

attached n-alkyl or semiperﬂuoroalkyl chains, form the ﬁrst “single plumber’s nightmare network”, the simplest soft-matter cubic

phase (Pm3m). Its cage-like grid comprises bundles of aromatic rods lying along the cubic unit cell edges, connected by six-way

hydrogen-bonded junctions. Side-chains ﬁll the remaining volume of this unique noninterpenetrating liquid-crystalline organic

framework.

icontinuous cubic (Cub ) phases of amphiphile−solvent

increased chain length is reﬂected in the expansion of the dV/

systems have found promising applications in biochem-

dr curve for SD compared to that for DG (Figure 1g). Figure

1g also shows how for double-network phases (DP, DD, DG)

dV/dr drops abruptly when the increasing radius r of the

network segments results in collision with the second network.

Meanwhile, in the absence of the second network, in SP, SG,

and SD, side-chains can be further extended until colliding

with parts of their own network. Figure 1g also implies that

reducing the side-chain volume should lead to a transition

from DG to DD and, ultimately, to the densest of all networks,

istry, biotechnology, and nanotechnology. They comprise

continua of two chemically diﬀerent constitutions. Most

prominent are the double-gyroid (DG, Ia3

DD, Pn3m), and double-network “plumber’s nightmare”

structures (DP, Im3

d), double-diamond

(

m) (Figure 1a−c). Cub phases, with

their unique 3D structures and high surface area, ﬁnd uses such

as templates for porous inorganic solids and membranes with

well-deﬁned nanopores.

^C^u^bbi phases also appear in block copolymers and

the “plumber’s nightmare”-type DP and SP phases (Im3

m and

thermotropic (solvent-free) liquid crystals (LC). In the latter,

the DG is predominant, DD is rare, and DP (Figure 1c)

Pm3m). Similarly, coarse-grain simulations predicted the SP in

T-shaped bolaamphiphiles beside the SD for smaller side-

−8

nonexistent. Beside these double networks a triple network

chains. While the DP is known in lyotropics, the SP has not

has also been reported, while biophotonic Cub_b_istructures

been experimentally observed in any system.

with only one network were found in butterﬂies (single gyroid,

From Figure 1g we can deduce a molecular design concept

for the envisaged SP phase. Ideally, normalized to the same

interjunction or bundle length L, the combined volume of the

side-chains should be signiﬁcantly smaller than required for the

DG or SD. Note that L = 1 in Figure 1g, which, for the SP

structure, is the unit cell length a. Meanwhile, at least one of

the chains must be long enough to reach from the network

segment central axis to r = 0.7L, which is where the dV/dr for

SP falls to 0. Furthermore, the dV/dr curves for SP and DP

show the largest downward departure from the straight line

dV/dr = 2πr, which is due to the large overlaps of the widening

cylinders at 6-fold junctions. This means that we should avoid

excessive fanning-out of the side-chain, and for this reason

branched chains should be avoided; note that the limiting case

10,11

SG) and beetles (single diamond, SD).

However,

synthetic versions have been obtained only by top-down

1,12

fabrication.

Generation of single-network structures

through self-assembly would simplify production of photonic

and metamaterials and of extremely porous materials, as the

second network is removed.

Recently the ﬁrst self-assembled single-network LC, the SD

with Fd3

m symmetry (Figure 1e), has been reported for a side-

branched glycerol-terminated p-terphenyl (compounds 1 with

n = 18−22, Figure 2a, right). This cubic LC belongs to the

so-called rod-bundle phases formed by bundles of rod-like

bolaamphiphiles with a laterally attached branched space-ﬁlling

14−17

chain (Figure 2a).

The bundles lie along the network

segments, and in the SD, as in the previously reported DD

of dV/dr = 2πr applies to uniaxial columnar phases, which

(

Pn3m), the segments are interconnected by tetrahedral

require highly divergent chains.

four-way junctions.

The number of consecutive bundles between junctions is

8,21

13,22

either two

or only one.

The observed replacement of

Received: October 21, 2019

Published: February 5, 2020

DG by SD phase (both with single bundles) upon increasing

the side-chain length of 1 from n = 16 to 18 (Figure 2a) was

attributed to the larger distance between network segments in

the SD with the same interjunction segment length. This

2020 American Chemical Society

Journal of the American Chemical Society

J. Am. Chem. Soc. 2020, 142, 3296−3300

296

The synthesis of 2 and 3 by C −C coupling reactions is

Communication

Figure 1. (a−c) Double-network cubic phases with their inﬁnite

minimal surfaces (blue) and (d−f) the corresponding single networks;

Figure 2. (a) Compounds 1 forming the double-gyroid (DG) and the

single-diamond (SD) cubic LC phases, depending on chain length n,

with schematic phase structures. (b) Formulas and molecular

(

g) radial volume distribution dV/dr curves, where V(r) is that part of

the unit cell volume that is within a distance r from the closest

network segment. The curves show the increase in occupied volume

as the radius (r) of the cylindrical network segments increases (the

dimensions of the new compounds 2 and 3; the volumes were

19,20

estimated using crystal volume increments.

length of the segments is normalized to 1).

Table 1. Transition Temperatures (T/°C), Associated

−

Enthalpy Changes (ΔH/kJ mol ), and Lattice Parameters

Therefore, we designed two diﬀerent molecules (Figure 2b).

In compound 2 the terphenyl was retained, but the long

branched chain was replaced by inward-pointing shorter ones

on the outer benzene rings. This led to a reduced fanning-out

of the chains. In this short-chain compound, semiperﬂuori-

nated chains were chosen to enhance nanosegregation to

ensure LC phase stability. In compound 3 we extended the

rod-like core of 2 by two additional ethynyl units and

replaced the short side-chain by two appropriately extended

linear n-alkyls attached to the middle ring. This increased the

bundle length L, which provided more internetwork space for

the longer side-chains to ﬁll.

of Compounds 2 and 3

a_c_u_b/nm (T/

compd

T/°C [ΔH/kJ·mol⁻¹]^a

H: Cr 122 [26.7] Cub/Pm3m 154

°C)

bundle

cell

2.89 (150)

19.3

6.4

[

2.7] Iso

C: Iso 151 [−2.6] Cub/Pm3

[−15.4] Cr

H: Cr 98 [74.8] Cub/Pm3

1.0] Iso

C: Iso 130 [−1.4] Cub/Pm3

[−78.0] Cr

m 44

m 134

3.61 (100)

26.5

8.8

[

m 79

−1

Determined by DSC (second heating/cooling, 10 K min , peak

temperatures); abbreviations: Cr = crystalline, Cub/Pm3m = simple

cubic phase (SP), Iso = isotropic liquid, H = heating, C = cooling, for

DSCs see Figure S1 .

described in Schemes S1 and S2 of the Supporting Information

SI), where the experimental procedures, analysis, and

simulation of diﬀraction patterns and additional data are also

given.

Pm3

possible by GISAXS on oriented ﬁlms (Figures 3b and S3a).

For the Im3m lattice the fundamental reﬂection group would

have been {110}, thus showing two diﬀraction spots in

addition to the one meridional and two equatorial; see Figure

S3b. Their absence means that the reﬂections belong to the

̅ ̅

m or a body-centered Im3m lattice. The distinction was

For both compounds cooling from the isotropic liquid (Iso)

induces an exothermic transition (ΔH = −1.4 to −2.6 kJ

−

mol ) (Figure S1) accompanied by a signiﬁcant increase in

viscosity, while the sample remains optically isotropic. The LC

character is conﬁrmed by the presence of only diﬀuse

scattering, the maximum being at d = 0.54 nm for compound

{100} group of the simple cubic lattice, excluding Im3m.

and at 0.45 nm for 3 (Figure S2). At smaller angles the

The cubic lattice parameters a_c_u_bof 2 and 3 are 2.89 and

3.61 nm, while the lengths L_m_o_lof their molecules are 2.5 and

3.1 nm (Figure 2b). The similarity of a and L_m_o_lsuggests a

structure in which bundles of coaxial molecular cores lie along

presence of sharp Bragg reﬂections indicates a periodic lattice.

For both 2 and 3 (Figure 3a,e) their 1/d values stand in the

ratio 1:√2:√3:√4:√5..., which can be indexed to a primitive

297

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J. Am. Chem. Soc. 2020, 142, 3296−3300

Communication

Figure 3. Cub/Pm3m phases of compounds (a−d) 2 and (e−h) 3 at the indicated temperatures: (a, e) experimental (top) and simulated (bottom)

SAXS powder diﬀractograms; (b) experimental GISAXS pattern (sharp spots are from well-oriented domains, lying on their {100} face); (c, g) ED

maps, blue-purple = high-density, red = low-density; schematic molecules are added; (f) model of 2 × 2 × 2 unit cells; (d, h) geometric models

used to calculate diﬀraction intensities shown in (a) and (e) and in Tables S3 and S4 ; mathematical details are in Section S3.2 .

the edges of the cubic unit cell, forming a simple cubic

framework with polar glycerol nodes, while the side-chains ﬁll

by the ﬂuoroalkyl segments of the side-chains (Figure 3c). In

the geometric model this is represented by the network of

fused body-centered octahedra (Figure 3d). Aromatic rod-

bundles together with the attached alkyl segments are now the

lowest ED moieties (red), while the glycerol cell corners are

intermediate (blue). One might expect that the map for 2

would be essentially the inverse of that for 3. However, in 2 the

body center is not part of the highest density region, although

the only moieties capable of reaching it are the high-ED

perﬂuoroalkyl segments. In fact, it turns out that the length l of

the fully stretched side-chains of 2, including the 0.45 nm core

width, is 1.85 nm, falling short of the required distance to the

body center ((a /2) × √2 = 2.04 nm). In other words l/a

= 1.85/2.89 = 0.64, i.e., slightly less than √2/2 = 0.7, where

the dV/dr for SP falls to 0 (Figure 1g). This shortfall in length

of the side-chains in 2 explains the local ED minimum at the

center of the unit cell (Figures 3c and S4c) and contributes to

the larger number of molecules in the bundles. No such

shortfall exists in 3, where l is 2.70 + 0.45 = 3.15 nm; hence l/

a_c_u_b= 3.15/3.61 = 0.87, well above the 0.7 threshold.

the remaining continuum (Figure 3c,f,g). The number of

molecules of volume V_m_o_lin a unit cell is n = 0.89a_c_u_b/V_m_o_l

cell

19.3 for 2, n = 26.5 for 3, and the factor 0.89 is a

correction (Table S2). Each unit cell contains three network

segments, each containing one bundle of 6 (6.4) molecules in

cell

and about 9 (8.8) in 3. The excess cell length a-L_m_o_lof 0.4−

.5 nm is attributed to the excluded volume eﬀect at the six-

way junctions (Figure 3f). Thus, in 2, 6n_b_u_n_d_l_e= 6 × 6.4 ≈ 38

glycerols aggregate in each H-bonded junction, adding to the

eﬀective bundle length. In 3 n_b_u_n_d_l_e= 9; hence their 54-glycerol

micelles cause an even larger excess cell length of 0.5 nm.

Geometric models of the cubic structures of 2 and 3 were

constructed (Figure 3d,h), and the diﬀraction intensities were

calculated, which compare well with those observed (Figure

cub

a,e). Next, electron density (ED) maps were reconstructed

using the structure factor amplitudes from diﬀraction

intensities and the corresponding phase angles from Fourier

analysis of the models (for details see Section S3.1 ).

In the ED map of compound 3 (Figures 3g and S4b) the

high-ED regions (blue-purple) follow the edges of the unit

cells forming a network of orthogonal columns with octahedral

As explained in Figure 4, the proposed structure of the Cub/

Pm3m phase of 2 was further corroborated by high-resolution

atomic force microscopy (AFM). The large domain size seen

by AFM is in line with the 0.185 μm size (lower limit)

obtained by the Scherrer equation from the synchrotron Bragg

intersections; this network contains the

1,4-bis-

(

phenylethynyl)benzene cores and the glycerols, as expected

−

for SP structure (Figure 3f). The low-ED alkyls (enclosed

within the red surface) occupy the space between the blue

network segments, around the areas that would be hosting the

second network in a DP phase. Although Figure 3g may be

reminiscent of a common depiction of the DP phase, as seen in

lyotropic LCs, note that in that case the two networks would

be the same color, while the minimum surface would be

diﬀraction line width of 0.034 nm .

In summary, the ﬁrst single “plumber’s nightmare” (SP)

bicontinuous network phase was discovered in two soft-matter

systems. It is only the second observed single-network LC

phase and is formed by self-assembly of strategically designed

π-shaped bolapolyphiles, the design being based on matching

the molecular architecture and phase geometry via the dV/dr

intercalated between them.

diagram. It belongs to the cubic bundle phases, being

In the ED map of compound 2 (Figures 3c and S4a) the

highest ED regions (blue-purple) are now the face centers

stretching toward the body center of the unit cell, dominated

bicontinuous regarding the network and micellar regarding

the glycerol spheres. This concept can lead to yet unknown LC

phases, including the still missing SG and DP (an “alternating”

298

Journal of the American Chemical Society

J. Am. Chem. Soc. 2020, 142, 3296−3300

orcid.org/0000-0002-4163-3272 ;

Communication

Goran Ungar − State Key Laboratory for Mechanical Behaviour

of Materials, Shaanxi International Research Center for Soft

Matter, Xi’an Jiaotong University, Xi’an 710049, People’s

Republic of China; Department of Materials Science and

Engineering, University of She ﬃ eld, Sheﬃeld S1 3JD, U.K.;

orcid.org/0000-0002-9743-2656; Email: g.ungar@

she ﬃ eld.ac.uk

Carsten Tschierske − Institute of Chemistry, Martin-Luther-

University Halle-Wittenberg, 06120 Halle, Germany;

Email: carsten.tschierske@chemie.uni-halle.de

Authors

Silvio Poppe − Institute of Chemistry, Martin-Luther-University

Halle-Wittenberg, 06120 Halle, Germany

Xiaohong Cheng − Institute of Chemistry, Martin-Luther-

University Halle-Wittenberg, 06120 Halle, Germany; Key

Laboratory of Medicinal Chemistry for Natural Resources,

Chemistry Department, Yunnan University, Kunming 650091,

People’s Republic of China

Changlong Chen − State Key Laboratory for Mechanical

Behaviour of Materials, Shaanxi International Research Center

for Soft Matter, Xi’an Jiaotong University, Xi’an 710049,

People’s Republic of China

Xiangbing Zeng − Department of Materials Science and

Engineering, University of She ﬃ eld, Sheﬃeld S1 3JD, U.K.;

orcid.org/0000-0003-4896-8080

Figure 4. (a) AFM phase image of 2 in the Cub/Pm3

m phase. Inset:

Fourier transform. (b) Fourier ﬁltered image in (a); the dark spots are

the stiﬀ ﬂuoroalkyl regions. (c) Six unit cells showing the highest

density aggregates of ﬂuorinated chains (compare Figure S4c) as

spheres around the face centers within an a_c_u_b/√2 = 2 nm thick

Rui-bin Zhang − Department of Materials Science and

Engineering, University of Sheﬃeld, Sheﬃeld S1 3JD, U.K.

(

110) plane (yellow box). (d) Frontal view of the cleaved (110)

surface with the content of the box in (c), showing the

pseudohexagonal arrangement of the ﬂuoroalkyl stiﬀness maxima,

seen in (b), where the white rectangle is an example area shown in (c)

and (d). XRD-based distances between rows (d) match closely those

measured by AFM (a).

Complete contact information is available at:

https://pubs.acs.org/10.1021/jacs.9b11073

Notes

The authors declare no competing ﬁnancial interest.

ACKNOWLEDGMENTS

DG has just been reported ). These unique LCs could be

considered as liquid organic frameworks (LOFs), related to the

■

This work is supported by the Deutsche Forschungsgemein-

schaft (392435074), the National Natural Science Foundation

of China (No. 21761132033, 21374086), the 111 Project 2.0

of China (BP2018008), and the Engineering and Science

Research Council UK (EP-P002250). For help with synchro-

tron experiments the authors thank the staﬀ at beamlines

BL16B1 at Shanghai Synchrotron Radiation Facility, I22 at

Diamond Light Source, UK, and BM28 (“XMaS”) at the

European Synchrotron Radiation Facility, France.

solid-state covalent and metal−organic frameworks (COFs,

MOFs ). The dynamic hydrogen-bonded junctions replace

the metal or covalent nodes of MOFs and COFs. The cage-like

noninterpenetrating grid of the Cub/Pm3m, reminiscent of

MOFs, is unknown among bicontinuous LCs. If scaled up,

these single-network structures are of interest for photonics

and metamaterials.

ASSOCIATED CONTENT

sı Supporting Information

The Supporting Information is available free of charge at

https://pubs.acs.org/doi/10.1021/jacs.9b11073.

■

REFERENCES

■

(1) Seddon, J. M.; Templer, R. H. Polymorphism in Lipid-Water

Systems. InHandbook of Biological Physics; Lipowsky, R.; Sackmann,

E., Eds.; Elsevier: Amsterdam, 1995; Vol. 1, pp 97−160.

Experimental methods; additional experimental data;

analysis of X-ray data, construction of maps and

simulation from models; synthesis and analytical data;

supporting discussion and calculations; additional NMR

spectra (PDF)

(

2) Meikle, T. G.; Drummond, C. J.; Separovic, F.; Conn, C. E.

Membrane-Mimetic Inverse Bicontinuous Cubic Phase Systems for

Encapsulation of Peptides and Proteins. In Advances in Biomembranes

and Lipid Self-Assembly; Iglic,

Academic Press, 2017; Vol. 25, pp 63−94.

3) Bicontinuous Liquid Crystals; Lynch, M. L.; Spicer, P. T., Eds.;

A.; Garcia-Saez, A.; Rappolt, M., Eds.;

̌ ́

(

Surfactant Science Series 127; CRC Press - Taylor & Francis Group:

Boca Raton, USA, 2005.

AUTHOR INFORMATION

Corresponding Authors

Feng Liu − State Key Laboratory for Mechanical Behaviour of

Materials, Shaanxi International Research Center for Soft

Matter, Xi’an Jiaotong University, Xi ’an 710049, People ’s

Republic of China; orcid.org/0000-0001-6224-5167;

Email: feng.liu@xjtu.edu.cn

■

(4) Kato, T.; Uchida, J.; Ichikawa, T.; Sakamoto, T. Functional

Liquid Crystals towards the Next Generation of Materials. Angew.

Chem., Int. Ed. 2018, 57 (16), 4355−4371.

(5) Meuler, A. J.; Hillmyer, M. A.; Bates, F. S. Ordered Network

Mesostructures in Block Polymer Materials. Macromolecules 2009, 42

(19), 7221−7250.

299

J. Am. Chem. Soc. 2020, 142, 3296−3300

Journal of the American Chemical Society

Communication

6) Kutsumizu, S. Recent Progress in the Synthesis and Structural

Clarification of Thermotropic Cubic Phases. Isr. J. Chem. 2012, 52

10), 844−853.

7) Zeng, X.; Ungar, G.; Imper

tricontinuous cubic liquid crystal. Nat. Mater. 2005, 4 (7), 562−567.

8) Dressel, C.; Liu, F.; Prehm, M.; Zeng, X.; Ungar, G.; Tschierske,

in Square and Triangular Pores. Adv. Funct. Mater. 2019, 29 (3),

1806078.

(

(26) These pi-shaped molecules can be considered as bolaamphi-

philes with additional lateral chains or alternatively as a special kind of

gemini amphiphiles with a rigid anisometric connection between the

two head groups; see for example: Sorenson, G. P.; Mahanthappa, M.

K. Soft Matter 2016, 12, 2408−2415.

(27) Chen, C.; Kieﬀer, R.; Ebert, H.; Prehm, M.; Zhang, R.-b.; Zeng,

X.; Liu, F.; Ungar, G.; Tschierske, C. Chirality Induction through

Nano-Phase Separation: Alternating Network Gyroid Phase by

Thermotropic Self-Assembly of X-Shaped Bolapolyphiles. Angew.

Chem., Int. Ed. 2020, 59, 2725−2729.

(28) Huang, N.; Wang, P.; Jiang, D. Covalent organic frameworks: a

materials platform for structural and functional designs. Nat. Rev.

Mater. 2016, 1, 16068.

or-Clerc, M. A triple-network

C. Dynamic Mirror-Symmetry Breaking in Bicontinuous Cubic

Phases. Angew. Chem., Int. Ed. 2014, 53 (48), 13115−13120.

(

Narayanan, S.; Sandy, A.; Dufresne, E. R.; Prum, R. O. Structure,

function, and self-assembly of single network gyroid (I 4 32) photonic

crystals in butterfly wing scales. Proc. Natl. Acad. Sci. U. S. A. 2010,

9) Saranathan, V.; Osuji, C. O.; Mochrie, S. G. J.; Noh, H.;

10) Wilts, B. D.; Michielsen, K.; De Raedt, H.; Stavenga, D. G.

07 (26), 11676−11681.

Hemispherical Brillouin zone imaging of a diamond-type biological

photonic crystal. J. R. Soc., Interface 2012, 9 (72), 1609−1614.

(29) O ’Keeffe, M.; Yaghi, O. M. Deconstructing the Crystal

Structures of Metal-Organic Frameworks and Related Materials into

Their Underlying Nets. Chem. Rev. 2012, 112 (2), 675−702.

11) Maldovan, M.; Urbas, A. M.; Yufa, N.; Carter, W. C.; Thomas,

E. L. Photonic properties of bicontinuous cubic microphases. Phys.

Rev. B: Condens. Matter Mater. Phys. 2002 , 65 , 165123.

(

30) Kanie, K.; Matsubara, M.; Zeng, X.; Liu, F.; Ungar, G.;

Nakamura, H.; Muramatsu, A. Simple cubic packing of gold

nanoparticles through rational design of their dendrimeric corona. J.

Am. Chem. Soc. 2012, 134 (2), 808−811.

(12) Han, L.; Che, S. An Overview of Materials with Triply Periodic

Minimal Surfaces and Related Geometry: From Biological Structures

to Self-Assembled Systems. Adv. Mater. 2018, 30 (17), 1705708.

(13) Zeng, X.; Poppe, S.; Lehmann, A.; Prehm, M.; Chen, C.; Liu,

F.; Lu, H.; Ungar, G.; Tschierske, C. A Self-Assembled Bicontinuous

Cubic Phase with a Single-Diamond Network. Angew. Chem., Int. Ed.

14) Tschierske, C. Liquid crystal engineering - new complex

019, 58 (22), 7375−7379.

mesophase structures and their relations to polymer morphologies,

nanoscale patterning and crystal engineering. Chem. Soc. Rev. 2007, 36

(

12), 1930−1970.

(

15) Tschierske, C.; Nurnberger, C.; Ebert, H.; Glettner, B.; Prehm,

M.; Liu, F.; Zeng, X.-B.; Ungar, G. Complex tiling patterns in liquid

crystals. Interface Focus 2012 , 2 (5), 669 −680.

(16) Tschierske, C. Development of Structural Complexity by

Liquid-Crystal Self-assembly. Angew. Chem., Int. Ed. 2013, 52 (34),

828−8878.

17) Prehm, M.; Liu, F.; Zeng, X.; Ungar, G.; Tschierske, C. Axial-

bundle phases − new modes of 2D, 3D, and helical columnar self-

assembly in liquid crystalline phases of bolaamphiphiles with swallow

tail lateral chains. J. Am. Chem. Soc. 2011, 133 (13), 4906−4916.

(18) Zeng, X.; Prehm, M.; Ungar, G.; Tschierske, C.; Liu, F.

Formation of a Double Diamond Cubic Phase by Thermotropic

Liquid Crystalline Self-Assembly of Bundled Bolaamphiphiles. Angew.

Chem., Int. Ed. 2016, 55 (29), 8324 −8327.

(19) Immirzi, A.; Perini, B. Prediction of Density in Organic

Crystals. Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen.

Crystallogr. 1977, 33 (1), 216−218.

(20) The factor 0.89 is a correction for the packing density of the LC

state related to the crystalline, assuming an average of the packing

density of between the liquid (0.55) and in the crystalline state (0.70).

The calculated packing density would be (0.55 + 0.70)/2 = 0.625,

corresponding to 0.89 times the packing density in the crystalline state

(

0.625/0.70 = 0.89).

(21) Liu, F.; Prehm, M.; Zeng, X.; Tschierske, C.; Ungar, G. Skeletal

Cubic, Lamellar, and Ribbon Phases of Bundled Thermotropic

Bolapolyphiles. J. Am. Chem. Soc. 2014, 136 (19), 6846 −6849.

(22) Poppe, S.; Chen, C.; Liu, F.; Tschierske, C. A skeletal double

gyroid formed by single coaxial bundles of catechol based

bolapolyphiles. Chem. Commun. 2018, 54 (79), 11196−11199.

(23) Sun, Y.; Padmanabhan, P.; Misra, M.; Escobedo, F. A.

Molecular dynamics simulation of thermotropic bolaamphiphiles with

a swallow-tail lateral chain: formation of cubic network phases. Soft

Matter 2017, 13 (45), 8542−8555.

(24) Glettner, B.; Liu, F.; Zeng, X.; Prehm, M.; Baumeister, U.;

Walker, M.; Bates, M. A.; Boesecke, P.; Ungar, G.; Tschierske, C.

Liquid-Crystalline Kagome. Angew. Chem., Int. Ed. 2008, 47 (47),

(

063−9066.

25) Zhang, R. B.; Zeng, X. B.; Wu, C. Y.; Jin, Q. H.; Liu, Y. S.;

Ungar, G. Square and Hexagonal Columnar Liquid Crystals Confined

300