Dynamic Mirror-Symmetry Breaking in Bicontinuous Cubic Phases**

Chiral segregation of enantiomers or chiral conformers of achiral molecules during self-assembly in well-ordered crystalline superstructures has fascinated chemists since Pasteur. Here we report spontaneous mirror-symmetry breaking in cubic phases formed by achiral multichain-terminated diphenyl-2,2′-bithiophenes. It was found that stochastic symmetry breaking is a general phenomenon observed in bicontinuous cubic liquid crystal phases of achiral rod-like compounds. In all compounds studied the ${{\it Im}\bar 3m}$ cubic phase is always chiral, while the ${Ia\bar 3d}$ phase is achiral. These intriguing observations are explained by propagation of homochiral helical twist across the entire networks through helix matching at network junctions. In the ${Ia\bar 3d}$ phase the opposing chiralities of the two networks cancel, but not so in the three-networks ${{\it Im}\bar 3m}$ phase. The high twist in the ${{\it Im}\bar 3m}$ phase explains its previously unrecognized chirality, as well as the origin of this complex structure and the transitions between the different cubic phases.


Synthesis and analytical data of the bithiophenes 1
The compounds 1 were synthesized in analogy to procedure 1.1.1 starting from the corresponding benzoic acid.

DSC-Investigations
Differential scanning calorimetry was done using a Perkin Elmer DSC-7 instrument. Typical heating and cooling rates were 10 K min -1 . Figure S1. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 1a in the temperature range between 105 and 170 °C. Figure S2. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 1b in the temperature range between 160 and 200 °C.

IsoHT
IsoHT Cr heating cooling S11 Figure S3. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 1c in the temperature range between 120 and 190 °C. Figure S4. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 1e in the temperature range between 160 and 190 °C.  Figure S5. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 1f in the temperature range between 93 and 139 °C. Figure S6. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 1g in the temperature range between 100 and 135 °C.  Figure S7. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 2 in the temperature range between 100 and 130 °C. Figure S8. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 3 in the temperature range between 135 and 160 °C. heating cooling S14 Figure S9. Section of the DSC heating and cooling scans (10 K min -1 ) of compound 4 in the temperature range between 140 and 170 °C.

CD and UV/VIS spectroscopy
UV/VIS spectrum was recorded on Lambda 14 (Perkin-Elmer). Microbeam circular dichroism (CD) spectroscopy experiments were performed at beamline B23 of the Diamond Light Source. An intense synchrotron-generated light beam of a fraction of a mm in diameter was used in the spectrometer, with the ability of samples being scanned in xy plane. This allowed only a small number, or even a single domain, to be captured, avoiding signal cancellation through spatial averaging. The beam was deflected vertically through the sample held between two quartz glass windows held in a Linkam hot stage.

X-ray scattering using laboratory source
X-ray investigations on powder-like samples were carried out with a Guinier film camera (Huber), samples in glass capillaries (1 mm) in a temperature-controlled heating stage, quartz-monochromatized CuK  radiation, 30 to 60 min exposure time, calibration with the powder pattern of Pb(NO3)2. Aligned samples were obtained on a glass plate. Alignment was achieved upon slow cooling (rate: 1 Kmin -1 -0.01 Kmin -1 ) of a small droplet of the sample and takes place at the sample-glass or at the sample-air interface, with domains fiber-like disordered around an axis perpendicular to the interface. The aligned samples were held on a temperature-controlled heating stage and the diffraction patterns were recorded with a 2D detector (HI-STAR, Siemens).
Small-and wide-angle X-ray experiments were also performed using a laboratory beamline based on a Xenocs Genix microfocus source with Fox2d single-bounce curved focusing multilayer optics and aBruker Vantec 2000 multiwire microgap gas detector.

Synchrotron X-ray diffraction and electron density reconstruction
High-resolution small-angle powder diffraction experiments were recorded on Beamline I22 at Diamond Light Source. Samples were held in evacuated 1 mm capillaries. A modified Linkam hot stage with a thermal stability within 0.2 ºC was used, with a hole for the capillary drilled through the silver heating block and mica windows attached to it on each side. A MarCCD detector was used. q calibration and linearization were verified using several orders of layer reflections from silver behemate and a series of n-alkanes. The measurement of the positions and intensities of the diffraction peaks is carried out using Galactic PeakSolve TM program, where experimental diffractograms are fitted using Gaussian shaped peaks. The diffraction peaks are indexed on the basis of their peak positions, and the lattice parameters and the space groups are subsequently determined. Once the diffraction intensities are measured and the corresponding space group determined, 3-d electron density maps can be reconstructed, on the basis of the general formula Here F(hkl) is the structure factor of a diffraction peak with index (hkl). It is normally a complex number and the experimentally observed diffraction intensity Here K is a constant related to the sample volume, incident beam intensity etc. In this paper we are only interested in the relative electron densities, hence this constant is simply taken to be 1. Thus the electron density

E(xyz) = hkl sqrt[I(hkl)] exp[i2π(hx+ky+lz)+hkl](Eqn. 3)
As the observed diffraction intensity I(hkl) is only related to the amplitude of the structure factor |F(hkl)|, the information about the phase of F(hkl), hkl, can not be determined directly from experiment. However, the problem is much simplified when the structure of the ordered phase is centrosymmetric, and hence the structure factor F(hkl) is always real and hkl is either 0 or π.

S16
This makes it possible for a trial-and-error approach, where candidate electron density maps are reconstructed for all possible phase combinations, and the "correct" phase combination is then selected on the merit of the maps, helped by prior physical and chemical knowledge of the system. This is especially useful for the study of nanostructures, where normally only a limited number of diffraction peaks are observed.
Grazing incidence small-angle (GISAXS) experiments were carried out on station BM28 (XMaS line) at European Synchrotron Radiation Facility (ESRF). Thin films were prepared from the melt on a silicon wafer. The thin film coated 5 x 5 mm 2 Si plates were placed on top of a custom built heater, which was then mounted on a six-circle goniometer. A MarCCD 165 detector was used. The sample enclosure and the beam pipe were flushed with helium.    Figure S15. Electron density maps of (a, b) the d Ia3 phase of compound 1c and (c, d) of the m Im 3 phase of compound 2 reconstructed from the data in the tables above. The isoelectron surfaces enclose the regions of highest electron density, i.e. the aromatic regions. The two

Electron density maps
nets are coloured differently, although the density levels are the same. In (b) the minimum surface is also shown in yellow. In (c) the three nets in (a) are coloured differently, although the density levels are the same. The "middle" network is yellow. This network is also shown separately in (d).
6. Additional details of structural models 6.1 Framework models with minimum surface Figure S16. Framework models of the d Ia3 (a) and m Im 3 (b) cubic phases, as in Fig. 1 but with the minimum surface added. In (a) the "gyroid" minimum surface separates the domains of the red and blue infinite networks. In (b) it closely follows the middle (yellow) network.

Calculation of number of molecules and the geometry of cubic networks
The total number of molecules N can be estimated from the lattice parameter acub (in nm), molar mass M(g mol -1 ) and assuming a density of 1.0 g cm -3 . The equation is N = 602.2 x acub 3 / M m Im 3 phase For 1e (acub = 17.87 nm and M =1184 g mol -1 ), N is 2.90 x 10 3 For 1g (acub = 15.36 nm and M = 1120 g mol -1 ), N is 1.95 x 10 3 The middle network contains 24 blue ribbons, 24 red ribbons and 24 green ribbons. The length of the blue ribbion is 0.152acub, and that of the red or green ribbon is 0.203acub, in total the length of the middle network is (0.152 + 2 x 0.203) x 24 x acub = 13.38 acub. The twisting angle of a blue ribbon is 180° and 240° for red and green ribbons.
The outer and inner networks each contain 12 edges of the octahedron and 3 links between the octahedra, the length of which is 0.325acub and 0.54acub respectively. The combined lengths of the outer and inner networks is thus (0.325 x 12 + 0.54 x 3) x 2 x acub = 11.05 acub.
The total length of all the segments in the unit cell is thus 24.43 acub. Assuming the distance between molecules along the segment is 0.45 nm, the number of "molecular layers" or strata a) b) is 54.29 acub. For 1e this is 970 layers, for 1g it is 834 layers. The number of molecules per stratum is thus 3.0 for 1e and 2.3 for 1g. The twist angle between successive strata is 180/[(0.152acub)/0.45] = 533/acub (degrees), which is 30° for 1e and 35° for 1g.

d Ia3 phase
There are a total of 24 segments in the d Ia3 unit cell, the length of each segment being 0.354acub. The total twist angle of each segment is 70.5°. The twist angle between successive molecular strata is then 70.5/[(0.354acub)/0.45] = 89.6/acub, which for 1b (acub = 11.40 nm) gives 7.9° and for 1a (acub = 10.84 nm) is 8.3°.
The total number of molecules per cell can be estimated as 844 for 1b, and 747 for 1a. The number of molecules in each segment is thus 35 and 31 respectively. As each segment contains 0.352/0.45 = 0.787acub molecule strata, the number of molecules per stratum is 3.9 (1b) and 3.6 (1a).

Optical microscopy
Polarized optical microscopy experiments were carried out on a Leica DMR XP in conjunction with a heating stage (FP 82 HT, Mettler) and controller (FP 90, Mettler).   Figure S22. Textures with chiral domains of compound 7 as observed between slightly decrossed polarizers (5°) after cooling from the isotropic at T = 55 °C; round boundaries of the chiral domains indicate the existence of an IsoLT [ * ] phase (not reported in [8]).