Mirror symmetry breaking in liquids and liquid crystals

References

• Guijarro A, Yus M. The origin of chirality in the molecules of life. Cambridge: RSC Publishing; 2009.
   Google Scholar
• Chirality in Liquid Crystals, ed. Kitzerow HS, Bahr, C. Springer, New York, 2001.
   Google Scholar
• Dierking I. Chiral liquid crystals: structures, phases, effects. Symmetry. 2014;6:444–472.
   Google Scholar
• Bisoyi HK, Bunning TJ, Li Q. Stimuli-driven control of the helical axis of self-organized soft helical superstructures. Adv Mater. 2018;1706512.
   Web of Science ®Google Scholar
• Khoo IC. Nonlinear optics, active plasmonics and metamaterials with liquid crystals. Prog Quant Electron. 2014;38:77–117.
   Web of Science ®Google Scholar
• http://bind.upatras.gr/
   Google Scholar
• Zehnacker A, Suhm MA. Chirality recognition between neutral molecules in the gas phase. Angew Chem Int Ed. 2008;47:6970–6992.
   PubMed Web of Science ®Google Scholar
• Tschierske C, Ungar G. Mirror symmetry breaking by chirality synchronisation in liquids and liquid crystals of achiral molecules. ChemPhysChem. 2016;17:9–26.
   PubMed Web of Science ®Google Scholar
• Pasteur L. Recherches sur les relations qui peuvent exister entre la forme crystalline, la composition chimique et le sens de la polarisation rotatoire. Ann Chim Phys. 1848;24:442459.
   Google Scholar
• Pérez-García L, Amabilino DB. Spontaneous resolution under supramolecular control. Chem Soc Rev. 2002;31:342–356.
   PubMed Web of Science ®Google Scholar
• Yashima E, Ousaka N, Taura D, et al. Supramolecular helical systems: helical assemblies of small molecules, foldamers, and polymers with chiral amplification and their functions. Chem Rev. 2016;116:13752−13990.
   PubMed Web of Science ®Google Scholar
• Liu M, Zhang L, Wang T. Supramolecular chirality in self-assembled systems. Chem Rev. 2015;115:7304–7397.
   PubMed Web of Science ®Google Scholar
• Barclay TG, Constantopoulos K, Matisons J. Nanotubes Self-Assembled from amphiphilic molecules via helical intermediates. Chem Rev. 2014;114:10217−10291.
   PubMed Web of Science ®Google Scholar
• Shimizu T, Musade M, Minamikawa H. Supramolecular nanotube architectures based on amphiphilic molecules. Chem Rev. 2005;105:1401–1443.
   PubMed Web of Science ®Google Scholar
• Fuhrhop J-H, Wang T. Bolaamphiphiles. Chem Rev. 2004;10:2901–2937.
   Web of Science ®Google Scholar
• Hoeben FJM, Jonkheijm P, Meijer EW, et al. About supramolecular assemblies of π-conjugated systems. Chem Rev. 2005;105:1491−1546.
   PubMed Web of Science ®Google Scholar
• Ribo JM, El-Hachemi Z, Arteaga O, et al. Hydrodynamic effects in soft-matter self-assembly: the case of j-aggregates of amphiphilic porphyrins. Chem Rec. 2017;17:713–724.
   PubMed Web of Science ®Google Scholar
• Selinger JV, Spector MS, Schnur JM. Theory of self-assembled tubules and helical ribbons. J Phys Chem. 2001;105:7157–7169.
   Web of Science ®Google Scholar
• Oda R, Huc I, Schmutz M, et al. Tuning bilayer twist using chiral counterions. Nature. 1999;399:566–569.
   PubMed Web of Science ®Google Scholar
• Palmans ARA, Meijer EW. Amplification of chirality in dynamic supramolecular aggregates. Angew Chem Int Ed. 2007;46:8948–8968.
   PubMed Web of Science ®Google Scholar
• Hill JP, Jin W, Kosaka A, et al. Self-assembled hexa-peri-hexabenzocoronene graphitic nanotube. Science. 2004;304:1481–1483.
   PubMed Web of Science ®Google Scholar
• Zhao M-Q, Zhang Q, Tian G-L, et al. Emerging double helical nanostructures. Nanoscale. 2014;6:9339–9354.
   PubMed Web of Science ®Google Scholar
• Snir Y, Kamien RD. Entropically driven helix formation. Science. 2005;307:1067.
   PubMed Web of Science ®Google Scholar
• Jinnai H, Kaneko T, Matsunaga K, et al. A double helical structure formed from an amorphous, achiral ABC triblock terpolymer. Soft Matter. 2009;5:2042–2046.
   Web of Science ®Google Scholar
• Green MM, Park J-W, Sato T, et al. The macromolecular route to chiral amplification. Angew Chem Int Ed. 1999;38:3138–3154.
   PubMed Web of Science ®Google Scholar
• Pijper D, Feringa BL. Control of dynamic helicity at the macro- and supramolecular level. Soft Matter. 2008;4:1349–1372.
   PubMed Web of Science ®Google Scholar
• Zhang L, Wang T, Shen Z, et al. Chiral nanoarchitectonics: towards the design, self-assembly, and function of nanoscale chiral twists and helices. Adv Mater. 2016;28:1044–1059.
   PubMed Web of Science ®Google Scholar
• Shen Z, Wang T, Liu M. Macroscopic chirality of supramolecular gels formed from achiral tris(ethyl cinnamate) benzene-1,3,5-tricarboxamides. Angew Chem Int Ed. 2014;53:13424–13428.
   PubMed Web of Science ®Google Scholar
• Duan P, Cao H, Zhang L, et al. Gelation induced supramolecular chirality: chirality transfer, amplification and application. Soft Matter, 2014;10:5428–5448.
   Google Scholar
• Zhang SY, Yang SY, Lan JB, et al. Helical nonracemic tubular coordination polymer gelators from simple achiral molecules. Chem Commun. 2008;6170–6172.
   PubMed Web of Science ®Google Scholar
• Wang D, Huang Y, Li J, et al. Lyotropic supramolecular helical columnar phases formed by C3- symmetric and unsymmetric rigid molecules. Chem Eur J. 2013;19:685–690.
   PubMed Web of Science ®Google Scholar
• Liu Y, Jia Y, Zhu E, et al. Supramolecular helical nanofibers formed by an achiral monopyrrolotetrathiafulvalene derivative: water-triggered gelation and chiral evolution. New J Chem. 2017;41:11060–11068.
   Web of Science ®Google Scholar
• Sekine T, Niori T, Watanabe J, et al. Spontaneous helix formation in smectic liquid crystals comprising achiral molecules. J Mater Chem. 1997;7:1307–1309.
   Web of Science ®Google Scholar
• Link DR, Natale G, Shao R, et al. Spontaneous formation of macroscopic chiral domains in a fluid smectic phase of achiral molecules. Science. 1997;278:1924−1927.
   PubMed Web of Science ®Google Scholar
• Reddy RA, Tschierske C. Bent-core liquid crystals: polar order, superstructural chirality and spontaneous desymmetrisation in soft matter systems. J Mater Chem. 2006;16:907–961.
   Web of Science ®Google Scholar
• Takezoe H. Spontaneous achiral symmetry breaking in liquid crystalline phases. In: Tschierske C, editor. Liquid crystals; Top Curr Chem. Vol. 318. Berlin: Springer; 2011. p. 303−330.
   Google Scholar
• Pelzl G, Diele S, Jakli A, et al. Helical superstructures in a novel smectic mesophase formed by achiral banana-shaped molecules. Liq Cryst. 2006;33:1519–1523.
   Web of Science ®Google Scholar
• Hough LE, Jung HT, Kruerke D, et al. Helical nanofilament phases. Science. 2009;325:456–460.
   PubMed Web of Science ®Google Scholar
• Le KV, Takezoe H, Araoka F. Chiral Superstructure mesophases of achiral bent-shaped molecules – hierarchical chirality amplification and physical properties. Adv Mater. 2017;1602737.
   Web of Science ®Google Scholar
• Zhu C, Wang C, Young A, et al. Probing and controlling liquid crystal helical nanofilaments. Nano Lett. 2015;15:3420–3424.
   PubMed Web of Science ®Google Scholar
• Otani T, Araoka F, Ishikawa K, et al. Enhanced optical activity by achiral rod-like molecules nanosegregated in the B4 structure of achiral bent-core molecules. J Am Chem Soc. 2009;131:12368–12372.
   PubMed Web of Science ®Google Scholar
• Chen D, Zhu C, Wang H, et al. Nanoconfinement of guest materials by helical nanofilament networks of bent-core mesogens. Soft Matter, 2013;9:462–471.
   Google Scholar
• Zep A, Salamonczyk M, Vaupotic N, et al. Physical gels made of liquid crystalline B4 phase. Chem Commun. 2013;49:3119–3121.
   Google Scholar
• Matraszek J, Topnani N, Vaupotic N, et al. Monolayer filaments versus multilayer stacking of bent-core molecules. Angew Chem Int Ed. 2016;55:3468–3472.
   PubMed Web of Science ®Google Scholar
• Choi S-W, Izumi T, Hoshino Y, et al. Circular-polarization-induced enantiomeric excess in an achiral bent-shaped liquid crystal. Angew Chem Int Ed. 2006;45:1382–1385.
   PubMed Web of Science ®Google Scholar
• Bialecka-Florianczyk E, Sledzinska I, Gorecka E, et al. Odd-even effect in biphenyl-based symmetrical dimers with methylene spacer - evidence of the B4 phase. Liq Cryst. 2008;35:401–406.
   Web of Science ®Google Scholar
• Sasaki H, Takanishi Y, Yamamoto J, et al. Achiral flexible liquid crystal trimers exhibiting gyroid-like surfaces in chiral conglomerate phases. Soft Matte. 2017;13:6521–6528.
   PubMed Web of Science ®Google Scholar
• Yoshizawa A, Kato Y, Sasaki H, et al. Optically isotropic homochiral structure produced by intercalation, of achiral liquid crystal trimers. J Phys Chem B. 2016;120:4843−4851.
   PubMed Web of Science ®Google Scholar
• Yoshizawa A. Molecular design of flexible liquid crystal oligomers stabilising the chiral frustrated phases. Liq Cryst. 2017;44:1877–1893.
   Web of Science ®Google Scholar
• Alaasar M, Prehm M, Brautzsch M, et al. Dark conglomerate phases of azobenzene derived bent-core mesogens – relationships between the molecular structure and mirror symmetry breaking in soft matter. Soft Matter. 2014;10:7285–7296.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Prehm M, Tschierske C. Helical nano-crystallite (HNC) phases: chirality synchronization of achiral bent-core mesogens in a new type of dark conglomerates. Chem Eur J. 2016;22:6583–6597.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Prehm M, Brautzsch M, et al. 4-Methylresorcinol based bent-core liquid crystals with azobenzene wings – a new class of compounds with dark conglomerate phases. J Mater Chem C. 2014;2:5487–5501.
   Web of Science ®Google Scholar
• Chen D, Shen Y, Aguero J, et al. Chiral isotropic sponge phase of hexatic smectic layers of achiral molecules. ChemPhysChem. 2014;15:1502–1507.
   PubMed Web of Science ®Google Scholar
• Tschierske C. Nanoscale stereochemistry in liquid crystals, in chirality at the nanoscale. Amabilino DB, editor. Weinheim,: Wiley-VCH Verlag GmbH & Co. KGaA; 2009. p. 271–304.
   Google Scholar
• Nagaraj M. Dark conglomerate phases of bent-core liquid crystals. Liq Cryst. 2016;43:2244–2253.
   Web of Science ®Google Scholar
• Dantlgraber G, Eremin A, Diele S, et al. Chirality and macroscopic polar order in a ferroelectric smectic liquid crystalline phase formed by achiral polyphilic bent-core molecules. Angew Chem Int Ed. 2002;41:2408–2412.
   PubMed Web of Science ®Google Scholar
• Nakata M, Link DR, Takanishi Y, et al. Takezoe, Electric-field-induced transition between the polarization-modulated and ferroelectric smectic-CSPF* liquid crystalline states studied using microbeam x-ray diffraction. Phy Rev E. 2005;71:011705.
   Web of Science ®Google Scholar
• Hough LE, Spannuth M, Nakata M, et al. Chiral isotropic liquids from achiral molecules. Science. 2009;325:452−456.
   PubMed Web of Science ®Google Scholar
• Ortega J, Folcia CL, Etxebarria J, et al. Electric-field-induced phase transitions in bent-core mesogens determined by x-ray diffraction. Phys Rev E. 2011;84:021707.
   Web of Science ®Google Scholar
• Deepa GB, Pratibha R. Chiral symmetry breaking dictated by electric-field-driven shape transitions of nucleating, conglomerate domains in a bent-core liquid crystal. Phys Rev E. 2014;89:042504.
   Web of Science ®Google Scholar
• Nagaraj M, Jones JC, Panov VP, et al. Understanding the unusual reorganization of the nanostructure of a dark conglomerate phase. Phys Rev E. 2015;91:04250413.
   Web of Science ®Google Scholar
• Eremin A, Diele S, Pelzl G, et al. Field-induced switching between states of opposite chirality in a liquid-crystalline phase. Phys Rev E. 2003;67:020702(R).
   Web of Science ®Google Scholar
• Pelzl G, Schröder MW, Eremin A, et al. Field-induced phase transitions and reversible field induced inversion of chirality in tilted smectic phases of bent-core mesogens. Eur Phys J E. 2006;21:293–303.
   PubMed Web of Science ®Google Scholar
• Deepa GB, Radhika S, Sadashiva BK, et al. Electric-field-induced switchable dark conglomerate phases in a bent-core liquid crystal exhibiting reverse columnar phases. Phys Rev E. 2013;87:062508.
   Web of Science ®Google Scholar
• Heppke G, Parghi DD, Sawade H. Novel sulphur-containing banana-shaped liquid crystal molecules. Liq Cryst. 2000;27:313–320.
   Web of Science ®Google Scholar
• Thisayukta J, Nakayama Y, Kawauchi S, et al. Distinct formation of a chiral smectic phase in achiral banana-shaped molecules with a central core based on a 2,7-dihydroxynaphthalene unit. J Am Chem Soc. 2000;122:7441–7448.
   Web of Science ®Google Scholar
• Zhang C, Diorio N, Lavrentovich OD, et al. Helical nanofilaments of bent-core liquid crystals with a second twist. Nat Commun. 2014;5:3302.
   PubMed Web of Science ®Google Scholar
• Dantlgraber G, Diele S, Tschierske C. The first liquid crystalline dimers consisting of two banana-shaped mesogenic units: a new way for switching between ferroelectricity and antiferroelectricity with bent-core molecules. Chem Commun. 2002;2768–2769.
   PubMed Web of Science ®Google Scholar
• Keith C, Reddy RA, Hauser A, et al. Silicon-containing polyphilic bent-core molecules: the importance of nanosegregation for the development of chirality and polar order in liquid crystalline phases formed by achiral molecules. J Am Chem Soc. 2006;128:3051–3066.
   PubMed Web of Science ®Google Scholar
• Reddy RA, Sadashiva BK. Ferroelectric properties exhibited by mesophases of compounds composed of achiral banana-shaped molecules. J Mater Chem. 2002;12:2627–2632.
   Web of Science ®Google Scholar
• Dantlgraber G, Baumeister U, Diele S, et al. Evidence for a new ferroelectric switching liquid crystalline phase formed by a carbosilane based dendrimer with banana-shaped mesogenic units. J Am Chem Soc. 2002;124:14852–14853.
   PubMed Web of Science ®Google Scholar
• Hahn H, Keith C, Lang H, et al. First example of a third-generation liquid-crystalline carbosilane dendrimer with peripheral bent-core mesogenic units: understanding of “dark conglomerate phases”. Adv Mater. 2006;18:2629–2633.
   Web of Science ®Google Scholar
• Keith C, Dantlgraber G, Reddy RA, et al. The influence of shape and size of silyl units on the properties of bent-core liquid crystals—from dimers via oligomers and dendrimers to polymers. J Mater Chem. 2007;17:3796–3805.
   Web of Science ®Google Scholar
• Keith C, Reddy RA, Tschierske C. The first example of a liquid crystalline side-chain polymer with bent-core mesogenic units: ferroelectric switching and spontaneous achiral symmetry breaking in an achiral polymer. Chem Commun. 2005;871–873.
   PubMed Web of Science ®Google Scholar
• Gimeno N, Sanchez-Ferrer A, Sebasti N, et al. Bent-core based main-chain polymers showing the dark conglomerate liquid crystal phase. Macromolecules. 2011;44:9586–9594.
   Web of Science ®Google Scholar
• Keith C, Reddy RA, Hahn H, et al. The carbosilane unit as a stable building block for liquid crystal design: a new class of ferroelectric switching banana-shaped mesogens. Chem Commun. 2004;1898–1899.
   PubMed Web of Science ®Google Scholar
• Zhang Y, Baumeister U, Tschierske C, et al. Achiral bent-core molecules with a series of linear or branched carbosilane termini: dark conglomerate phases, supramolecular chirality and macroscopic polar order. Chem Mater. 2010;22:2869.
   Web of Science ®Google Scholar
• Reddy RA, Zhu C, Shao R, et al. Spontaneous ferroelectric order in a bent-core smectic liquid crystal of fluid orthorhombic layers. Science. 2011;332:72–77.
   PubMed Web of Science ®Google Scholar
• Srigengan S, Nagaraj M, Ferrarini A, et al. Anomalously low twist and bend elastic constants in an oxadiazole-based bent-core nematic liquid crystal and its mixtures; contributions of spontaneous chirality and polarity. J Mater Chem C. 2018;6:980—988.
   Web of Science ®Google Scholar
• Etxebarria J, Folcia CL, Ortega J. Comment on “optical activity produced by layer chirality in bent-core liquid crystals. Phys Rev Lett. 2008;101:079801.
   PubMed Web of Science ®Google Scholar
• Hough LE, Zhu C, Nakata M, et al. Reply. Phys Rev Lett. 2008;101:079802.
   PubMed Web of Science ®Google Scholar
• Hough LE, Clark NA. Layer-scale optical chirality of liquid-crystalline phases. Phys Rev Lett. 2005;95:107802.
   PubMed Web of Science ®Google Scholar
• Jakli A, Huyng YM, Fodor-Csorba K, et al. Reversible switching between optically isotropic and birefringent statesin a bent-core liquid crystal. Adv Mater. 2003;15:1606–1610.
   Web of Science ®Google Scholar
• Görtz V, Goodby JW. Enantioselective segregation in achiral nematic liquid crystals. Chem Commun. 2005;3262.
   PubMed Web of Science ®Google Scholar
• Walba DM, Eshdat L, Körblova E, et al. On the nature of the B4 banana phase: crystal or not a crystal? Cryst Growth Des. 2005;5:2091.
   Web of Science ®Google Scholar
• Earl DJ, Osipov MA, Takezoe H, et al. Induced and spontaneous deracemization in bent-core liquid crystal phases and in other phases doped with bent-core molecules. Phys Rev E. 2005;71:021706.
   Web of Science ®Google Scholar
• Niwano H, Nakata M, Thisayukta J, et al. Chiral memory on transition between the B2 and B4 phases in an achiral banana-shaped molecular system. J Phys Chem B. 2004;108:14889.
   Web of Science ®Google Scholar
• Xu J, Dong RY, Domenici V, et al. 13C and 2H NMR study of structure and dynamics in banana B2 phase of a bent-core mesogen. J Phys Chem B. 2006;110:9434.
   PubMed Web of Science ®Google Scholar
• Dressel C, Reppe T, Prehm M, et al. Chiral self-sorting and amplification in isotropic liquids of achiral molecules. Nat Chem. 2014;6:971–977.
   PubMed Web of Science ®Google Scholar
• Ocak H, Bilgin-Eran B, Prehm M, et al. Effects of molecular chirality on superstructural chirality in liquid crystalline dark conglomerate phases. Soft Matter. 2012;8:7773–7783.
   Web of Science ®Google Scholar
• Liao G, Stojadinovic S, Pelzl G, et al. Optically isotropic liquid-crystal phase of bent-core molecules with polar nanostructure. Phys Rev E. 2005;72:021710.
   Web of Science ®Google Scholar
• Kauffman S. At home in the Universe. Offord: Oxford University Press; 1995. p. 54–58.
   Google Scholar
• Nagaraj M, Usami K, Zhang Z, et al. Unusual electric-field-induced transformations in the dark conglomerate phase of a bent-core liquid crystal. Liq Cryst. 2014;41:800–811.
   Web of Science ®Google Scholar
• Pelzl G, Eremin A, Diele S, et al. Spontaneous chiral ordering in the nematic phase of an achiral banana-shaped compound. J Mater Chem. 2002;12:2591−2593.
   Web of Science ®Google Scholar
• Alaasar M, Prehm M, Nagaraj M, et al. A liquid crystalline phase with uniform tilt, local polar order and capability of symmetry breaking. Adv Mater. 2013;25:2186–2191.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Prehm M, May K, et al. 4-Cyanoresorcinol-based bent-core mesogens with azobenzene wings: emergence of sterically stabilized polar order in liquid crystalline phases. Adv Funct Mater. 2014;24:1703–1717.
   Web of Science ®Google Scholar
• Nagayama H, Varshney SK, Goto MF, et al. Spontaneous deracemization of disc-like molecules in the columnar phase. Angew Chem Int Ed. 2010;49:445–448.
   PubMed Web of Science ®Google Scholar
• Kajitani T, Kohmoto S, Yamamoto M, et al. Spontaneous chiral induction in a cubic phase. Chem Mater. 2005;17:3812−3819.
   Web of Science ®Google Scholar
• Dressel C, Liu F, Prehm M, et al. Dynamic mirror-symmetry breaking in bicontinuous cubic phases. Angew Chem Int Ed. 2014;53:13115−13120.
   PubMed Web of Science ®Google Scholar
• Jeong HS, Tanaka S, Yoon DK, et al. Spontaneous chirality induction and enantiomer separation in liquid crystals, composed of achiral rod-shaped 4-arylbenzoate esters. J Am Chem Soc. 2009;131:15055−15060.
   PubMed Web of Science ®Google Scholar
• Kajitani T, Masu H, Kohmoto S, et al. Generation of a chiral mesophase by achiral molecules: absolute chiral induction in the smectic C phase of 4-octyloxyphenyl 4-octyloxybenzoate. J Am Chem Soc. 2005;127:1124–1125.
   PubMed Web of Science ®Google Scholar
• Kadkin ON, Kim EH, Rha YJ, et al. Novel tetrahedratic smectic C and nematic mesophases in unsymmetrically 1,1’-bis-substituted ferrocenomesogens. Chem Eur J. 2009;15:10343–10347.
   PubMed Web of Science ®Google Scholar
• Jeong K-U, Knapp BS, Ge JJ, et al. Origin of self-assembled helical supramolecular structures in achiral C6 biphenyl carboxylic acid compounds. Chem Mater. 2006;18:680–690.
   Web of Science ®Google Scholar
• Glogarova M, Hampl F, Lejček L, et al. Experimental proof of symmetry breaking in tilted smectics composed of molecules with axial chirality. J Chem Phys. 2010;133:221102.
   PubMed Web of Science ®Google Scholar
• Walba DM, Körblova E, Huang C-C, et al. Reflection symmetry breaking in achiral rod-shaped smectic liquid crystals? J Am Chem Soc. 2006;128:5318–5319.
   PubMed Web of Science ®Google Scholar
• Jeong K-U, Yang D-K, Graham M, et al. Construction of chiral propeller architectures from achiral molecules. Adv Mater. 2006;18:3229–3232.
   Web of Science ®Google Scholar
• Panov VP, Nagaraj M, Vij JK, et al. Periodic deformations in nonchiral planaraligned bimesogens with a nematic-nematic transition and a negative elastic constant. Phys Rev Lett. 2010;105:167801.
   PubMed Web of Science ®Google Scholar
• Torgova SI, Petrov MP, Strigazzi A. Textures of homologous 4-n-alkyloxybenzoic acids: spontaneous chirality and surface memory. Liq Cryst. 2001;28:1439–1449.
   Web of Science ®Google Scholar
• Young WR, Aviram A, Cox RJ. Stilbene derivatives. A new class of room temperature nematic liquids. J Am Chem Soc. 1972;94:3976–3981.
   Web of Science ®Google Scholar
• Yoshizawa A, Kato H. Coexistence of nematic and chiral nematic phases of an achiral liquid crystal trimer possessing an octafluorobiphenyl unit. Liqu Cryst. 2018;45:1443–1450.
   Web of Science ®Google Scholar
• Takanishi Y, Takezoe H, Suzuki Y, et al. Spontaneous enantiomeric resolution in a fluid smectic phase of a racemate. Angew Chem Int Ed. 1999;38:2353–2356.
   PubMed Web of Science ®Google Scholar
• Cowling SJ, Hall AW, Goodby JW. Electrooptical response in a racemic SmC liquid crystal. Adv Mater. 2005;17:1077–1080.
   Web of Science ®Google Scholar
• Kane A, Shao R-F, Maclennan JE, et al. Electric-field-driven deracemization. ChemPhysChem. 2007;8:170–174.
   PubMed Web of Science ®Google Scholar
• Baumgarten JL. Ferrochirality: a simple theoretical model of interacting, dynamically invertible, helical polymers, 2 a) molecular field approach: supports and the details. Macromol Theory Simul. 1995;4:1–43.
   Web of Science ®Google Scholar
• Toxvaerd S. Molecular dynamics simulations of isomerization kinetics in condensed fluids. Phys Rev Lett. 2000;85:4747–4750.
   PubMed Web of Science ®Google Scholar
• Latinwo F, Stillinger FH, Debenedetti PG. Molecular model for chirality phenomena. J Chem Phys. 2016;145:154503.
   PubMed Web of Science ®Google Scholar
• Salter PS, Benzie PW, Reddy RA, et al. Spontaneously chiral domains of an achiral bent-core nematic liquid crystal in a planar aligned device. Phys Rev E. 2009;80:031701.
   Web of Science ®Google Scholar
• Görtz V, Southern C, Roberts NW, et al. Unusual properties of a bent-core liquid-crystalline fluid. Soft Matter. 2009;5:463−471.
   Web of Science ®Google Scholar
• Keith C, Lehmann A, Baumeister U, et al. Nematic phases of bent-core mesogens. Soft Matter. 2010;6:1704–1721.
   Web of Science ®Google Scholar
• Francescangeli O, Vita F, Samulski ET. The cybotactic nematic phase of bent-core mesogens: state of the art and future developments. Soft Matter. 2014;10:7685–7691.
   PubMed Web of Science ®Google Scholar
• Rest C, Kandanelli R, Fernandez G. Strategies to create hierarchical self-assembled structures via cooperative non-covalent interactions. Chem Soc Rev. 2015;44:2543–2572.
   PubMed Web of Science ®Google Scholar
• Cantekin S, Ten Eikelder HMM, Markvoort AJ, et al. Consequences of cooperativity in racemizing supramolecular systems. Angew Chem Int Ed. 2012;51:6426–6431.
   PubMed Web of Science ®Google Scholar
• Mahadevi AS, Sastry GN. Cooperativity in noncovalent interactions. Chem Rev. 2016;116:2775−2825.
   PubMed Web of Science ®Google Scholar
• Pescitelli G, Di Bari L, Berova N. Application of electronic circular dichroism in the study of supramolecular systems. Chem Soc Rev. 2014;43:5211–5233.
   PubMed Web of Science ®Google Scholar
• Gottarelli G, Lena S, Masiero S, et al. The use of circular dichroism spectroscopy for studying the chiral molecular self-assembly: an overview. Chirality. 2008;20:471–485.
   PubMed Web of Science ®Google Scholar
• Shi Y, Sun Z, Chen R, et al. Effect of Conformational Chirality on Optical Activity Observed in a Smectic Phase of Achiral, Bent-Core Molecules. J Phys Chem B, 2017;121:6944–6950.
   Google Scholar
• Sreenilayam SP, Panarin YP, Vij JK, et al. Spontaneous helix formation in non-chiral bent-core liquid crystals with fast linear electro-optic effect. Nat Commun. 2016;7:11369.
   PubMed Web of Science ®Google Scholar
• Panarin YP, Sreenilayam SP, Vij JK, et al. A fast linear electro-optical effect in a non-chiral bent-core liquid crystal. J Mater Chem C. 2017;5:12585—12590.
   Web of Science ®Google Scholar
• Pikin SA, Indenbom VL. Piezoeffects and ferroelectric phenomena in smectic liquid crystals. Ferroelectrics. 1978;20:151–153.
   Web of Science ®Google Scholar
• Takanishi Y, Ohtsuka Y, Takahashi YS, et al. Chiral doping effect in the B2 phase of a bent-core liquid crystal: the observation of resonant X-ray satellite peaks assigned to the 5/10 layer periodic structure. Eur Phys Lett. 2015;109:56003.
   Google Scholar
• Abberley JP, Killah R, Walker R, et al. Heliconical smectic phases formed by achiral molecules. Nat Commun. 2018;9:228.
   PubMed Web of Science ®Google Scholar
• Merkel K, Kocot A, Vij JK, et al. Electric field distortions of the twist bend nematic (NTB) phase of bent-core liquid crystals.arXiv [cond-mat.soft]. 2018;1805.02936.
   Google Scholar
• Tschierske C, Photinos DJ. Biaxial nematic phases. J Mater Chem. 2010;20:4263–4294.
   Web of Science ®Google Scholar
• Berardi R, Muccioli L, Orlandi S, et al. Computer simulations of biaxial nematics. J Phys Cond Matter. 2008;20:463101.
   PubMed Web of Science ®Google Scholar
• Westphal E, Gallardo H, Finoto Caramori G, et al. Polar order and symmetry breaking at the boundary between bent-core and rodlike molecular forms: when 4-cyanoresorcinol meets the carbosilane end group. Chem Eur J. 2016;22:8181–8197.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Prehm M, Poppe S, et al. Development of polar order by liquid-crystal self-assembly of weakly bent molecules. Chem Eur J. 2017;23:5541–5556.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Prehm M, Tamba M-G, et al. Development of polar order in the liquid crystal phases of a 4-cyanoresorcino based bent-core mesogen with fluorinated azobenzene wings. ChemPhysChem. 2016;17:278–287.
   PubMed Web of Science ®Google Scholar
• Peroukidis SD, Vanakaras AG, Ph.otinos DJ. Molecular simulation of hierarchical structures in bent-core nematic liquid crystals. Phys Rev E. 2011;84:010702(R).
   Web of Science ®Google Scholar
• Mauguin C. Sur les cristaux liquides de Lehmann. Bull Soc Fr Mineral. 1911;34:71–117.
   Google Scholar
• Chen C-W, Li -C-C, Jau HC, et al. Bistable light-driven π phase switching using a twisted nematic liquid crystal film. Opt Expr. 2014;22:12133.
   Google Scholar
• Tortora L, Lavrentovich OD. Chiral symmetry breaking by spatial confinement in tactoidal droplets of lyotropic chromonic liquid crystals. Proc Natl Acad Sci U S A. 2011;108:5163−5168.
   PubMed Web of Science ®Google Scholar
• Dietrich CF, Rudquist R, Lorenz K, et al. Chiral structures from achiral micellar lyotropic liquid crystals under capillary confinement. Langmuir. 2017;33:5852−5862.
   PubMed Web of Science ®Google Scholar
• Basu R, Pendery JS, Petschek RG, et al. Macroscopic torsional strain and induced molecular conformational deracemization. Phys Rev Lett. 2011;107:237804.
   PubMed Web of Science ®Google Scholar
• Jo S-Y, Kim B-C, Jeon S-W, et al. Enhancement of the helical twisting power with increasing the terminal chain length of nonchiral bent-core molecules doped in a chiral nematic liquid crystal. RSC Adv. 2017;7:1932–1935.
   Web of Science ®Google Scholar
• Gorecka E, Cepic M, Mieczkowski J, et al. Enhanced chirality by adding achiral molecules into the chiral system. Phys Rev E. 2003;67:061704.
   Web of Science ®Google Scholar
• Weissflog W, Sokolowski S, Dehne H, et al. Chiral ordering in the nematic and an optically isotropic mesophase of bent-core mesogens with a halogen substituent at the central core. Liq Cryst. 2004;31:923–933.
   Web of Science ®Google Scholar
• Niori T, Yamamoto J, Yokoyama H. Dynamics of the nematic phase formed by achiral banana-shaped materials. Mol Cryst Liq Cryst. 2004;409:475–482.
   Web of Science ®Google Scholar
• Panov VP, Balachandran R, Vij JK, et al. Field-induced periodic chiral pattern in the Nx phase of achiral bimesogens. Appl Phys Lett. 2012;101:234106.
   Web of Science ®Google Scholar
• Cestari M, Diez-Berart S, Dunmur DA, et al. Phase behavior and properties of the liquid-crystal dimer 1′′,7′′-bis(4-cyanobiphenyl-4′-yl) heptane: a twist-bend nematic liquid crystal. Phys Rev E. 2011;84:031704.
   PubMed Web of Science ®Google Scholar
• Henderson PA, Imrie CT. Methylene-linked liquid crystal dimers and the twist–bend nematic phase. Liq Cryst. 2011;38:1407–1414.
   Web of Science ®Google Scholar
• Borshch V, Kim Y-K, Xiang J, et al. Nematic twist-bend phase with nanoscale modulation of molecular orientation. Nat Commun. 2013;4:2635.
   PubMed Web of Science ®Google Scholar
• Meyer RB. Structural problems in liquid crystal physics, in Les Houches summer school in theoretical physics. Balian R, Weil W, editors. New York: Gordon and Breach; 1976. p. 273.
   Google Scholar
• Dozov I. On the spontaneous symmetry breaking in the mesophases of achiral banana-shaped molecules. Europhys Lett. 2001;56:247–253.
   Google Scholar
• Memmer R. Liquid crystal phases of achiral banana-shaped molecules: a computer simulation study. Liq Cryst. 2002;29:483–496.
   Web of Science ®Google Scholar
• Stevenson WD, An J, Zeng X-B, et al. Twist-bend nematic phase in biphenylethanebased Copolyethers. Soft Matter. 2018;14:3003—3011.
   PubMed Web of Science ®Google Scholar
• Salamonczyk M, Vaupotic N, Pociecha D, et al. Structure of nanoscale-pitch helical phases: blue phase and twist-bend nematic phase resolved by resonant soft X-ray scattering. Soft Matter. 2017;13:6694—6699.
   PubMed Web of Science ®Google Scholar
• Emsley JW, Lelli M, Lesage A, et al. A comparison of the conformational distributions of the achiral symmetric liquid crystal dimer CB7CB in the achiral nematic and chiral twist-bend nematic phases. J Phys Chem B. 2013;117:6547–6557.
   PubMed Web of Science ®Google Scholar
• Hoffmann A, Vanakaras AG, Kohlmeier A, et al. On the structure of the Nx phase of symmetric dimers: inferences from NMR. Soft Matter. 2015;11:850–855.
   PubMed Web of Science ®Google Scholar
• Sepelj M, Lesac A, Baumeister U, et al. Dimeric Salicylaldimine-based mesogens with flexible spacers: parity-dependent mesomorphism. Chem Mater. 2006;18:2050–2058.
   Web of Science ®Google Scholar
• Schröder MW, Diele S, Pelzl G, et al. Different nematic phases and a switchable SmCP phase formed by homologues of a new class of asymmetric bent-core mesogens. J Mater Chem. 2003;13:1877–1882.
   Google Scholar
• Attard GS, Date RW, Imrie CT, et al. Non-symmetric dimeric liquid crystals the preparation and properties of the α-(4-cyanobiphenyl-4′-yloxy)-ω-(4- n-alkylanilinebenzylidene-4′-oxy)alkanes. Liq Cryst. 1994;16:529–581.
   Google Scholar
• Ungar G, Percec V, Zuber M. Liquid crystalline polyethers based on conformational isomerism. 20.1, nematic-nematic transition in polyethers and copolyethers based on l-(4-hydroxyphenyl)-2-(2-r-4-hydroxyphenyl)ethane with R = fluoro, chloro, and methyl and flexible spacers containing an odd number of methylene units. Macromolecules. 1992;25:75–80.
   Web of Science ®Google Scholar
• Panov VP, Vij JK, Mehl GH. Twist-bend nematic phase in cyanobiphenyls and difluoroterphenyls bimosogens. Liq Cryst. 2017;44:147–159.
   Web of Science ®Google Scholar
• Chen D, Porada JH, Hooper JB, et al. Chiral heliconical ground state of nanoscale pitch in a nematic liquid crystal of achiral molecular dimers. Proc Natl Acad Sci USA. 2013;110:15931–15936.
   PubMed Web of Science ®Google Scholar
• Chen D, Nakata M, Shao R, et al. Twist-bend heliconical chiral nematic liquid crystal phase of an achiral rigid bent-core mesogen. Phys Rev E. 2014;89:0225067.
   Google Scholar
• Zhu C, Tuchband MR, Young A, et al. Resonant carbon K-edge soft X-ray scattering from lattice-free heliconical molecular ordering: soft dilative elasticity of the twistbend liquid crystal phase. Phys Rev Lett. 2016;116:147803.
   Google Scholar
• Archbold CT, Davis EJ, Mandle RJ, et al. Chiral dopants and the twist-bend nematic phase—induction of novel mesomorphic behaviour in an apolar bimesogen. Soft Matter. 2015;11:7547–7557.
   Google Scholar
• Mandle RJ, Goodby JW. Intercalated soft-crystalline mesophase exhibited by an unsymmetrical twist-bend nematogen. CrystEngComm. 2016;18:8794–8802.
   Web of Science ®Google Scholar
• Mandle RJ, Goodby JW. A twist-bend nematic to an intercalated, anticlinic, biaxial phase transition in liquid crystal bimesogens. Soft Matter. 2016;12:1436–1443.
   PubMed Web of Science ®Google Scholar
• Sebastián N, Tamba MG, Stannarius R, et al. Mesophase structure and behaviour in bulk and restricted geometry of a dimeric compound exhibiting a nematic–nematic transition. Phys Chem Chem Phys. 2016;18::19299–19308.
   Web of Science ®Google Scholar
• Imrie CT, Henderson PA, Yeap G-Y. Liquid crystal oligomers: going beyond dimers. Liq Cryst. 2009;36:755–777.
   Web of Science ®Google Scholar
• Salili SM, Ribeiro De Almeida RR, Challa PK, et al. Spontaneously modulated chiral nematic structures of flexible bent-core liquid crystal dimers. Liq Cryst. 2017;44:160–167.
   Web of Science ®Google Scholar
• Meyer C, Luckhurst GR, Dozov I. The temperature dependence of the heliconical tilt angle in the twist-bend nematic phase of the odd dimer CB7CB. J Mater Chem C. 2015;3:318–328.
   Web of Science ®Google Scholar
• Mandle RJ. The shape of things to come: the formation of modulated nematic mesophases at various length scales. Chem Eur J. 2017;23:8771–8779.
   PubMed Web of Science ®Google Scholar
• Mandle RJ. The dependency of twist-bend nematic liquid crystals on molecular structure: a progression from dimers to trimers, oligomers and polymers. Soft Matter. 2016;12:7883—7901.
   PubMed Web of Science ®Google Scholar
• Mandle RJ. Designing liquid-crystalline oligomers to exhibit twist-bend modulated nematic phases. Chem Rec. 2018;18:1–10.
   Web of Science ®Google Scholar
• Goodby JW. Free volume, molecular grains, self-organisation, and anisotropic entropy: machining materials. Liq Cryst. 2017;44:1755–1763.
   Web of Science ®Google Scholar
• Paterson DA, Abberley JP, Harrison WT, et al. Cyanobiphenyl-based liquid crystal dimers and the twist-bend nematic phase. Liq Cryst. 2017;44:127–146.
   Web of Science ®Google Scholar
• Paterson DA, Gao M, Kim YK, et al. Understanding the twist-bend nematic phase: the characterisation of 1-(4-cyanobiphenyl-4 ‘-yloxy)-6-(4-cyanobiphenyl-4 ‘-yl)hexane (CB6OCB) and comparison with CB7CB. Soft Matter. 2016;12:6827–6840.
   PubMed Web of Science ®Google Scholar
• Sreenilayam SP, Panov VP, Vij JK, et al. The NTB phase in an achiral asymmetrical bent-core liquid crystal terminated with symmetric alkyl chains. Liqu Cryst. 2017;44:244–253.
   Web of Science ®Google Scholar
• Wang Y, Singh G, Agra-Kooijman DM, et al. Room temperature heliconical twist-bend nematic liquid crystal. CrystEngComm. 2015;17:2778–2782.
   Web of Science ®Google Scholar
• Simpson FP, Mandle RJ, Moore JN, et al. Investigating the cusp between the nano and macro-sciences in supermolecular liquid-crystalline twist-bend nematogens. J Mater Chem C. 2017;5:5102—5110.
   Web of Science ®Google Scholar
• Walker R, Pociecha D, Abberley JP, et al. Spontaneous chirality through mixing achiral components: a twist-bend nematic phase driven by hydrogen-bonding between unlike components. Chem Commun. 2018;54:3383—3386.
   PubMed Web of Science ®Google Scholar
• Pocock EE, Mandle RJ, Goodby JW. Molecular shape as a means to control the, incidence of the nanostructured twist bend phase. Soft Matter. 2018;14:2508—2514.
   PubMed Web of Science ®Google Scholar
• Dawood AA, Grossel MC, Luckhurst GR, et al. On the twist-bend nematic phase formed directly from the isotropic phase. Liq Cryst. 2016;43:2–12.
   Web of Science ®Google Scholar
• Dawood AA, Grossel MC, Luckhurst JR, et al. Twist-bend nematics, liquid crystal dimers, structure–property relations. Liq Cryst. 2017;44:106–126.
   Web of Science ®Google Scholar
• Salili SM, Kim C, Sprunt S, et al. Flow properties of a twist-bend nematic liquid crystal. RSC Adv. 2014;4:57419–57423.
   Web of Science ®Google Scholar
• Mandle RJ, Davis EJ, Archbold CT, et al. Microscopy studies of the nematic NTB phase of 1,11-di-(1”-cyanobiphenyl-4-yl)undecane. J Mater Chem C. 2014;2:556–566.
   PubMed Web of Science ®Google Scholar
• Tamba MG, Salili SM, Zhang C, et al. A fibre forming smectic twist–bent liquid crystalline phase. RSC Adv. 2015;5:11207–11211.
   Web of Science ®Google Scholar
• Mandle RJ, Goodby JW. Does topology dictate the incidence of the twist-bend phase? Insights gained from novel unsymmetrical bimesogens. Chem Eur J. 2016;22:18456–18464.
   PubMed Web of Science ®Google Scholar
• Meyer C, Luckhurst GR, Dozov I. Flexoelectrically driven electroclinic effect in the twist-bend nematic phase of achiral molecules with bent shapes. Phys Rev Lett. 2013;111:067801.
   PubMed Web of Science ®Google Scholar
• Greco C, Ferrarini A. Entropy-driven chiral order in a system of achiral bent particles. Phys Rev Lett. 2015;115:147801.
   PubMed Web of Science ®Google Scholar
• Stevenson WD, Ahmed Z, Zeng XB. Molecular organization in the twist–bend nematic phase by resonant X-ray scattering at the Se K-edge and by SAXS, WAXS and GIXRD. Phys Chem Chem Phys. 2017;19:13449.
   PubMed Web of Science ®Google Scholar
• Barry E, Hensel Z, Dogic Z, et al. R. Entropy driven formation of a chiral liquid crystalline phase of helical filaments. Phys Rev Lett. 2006;96:018305.
   PubMed Web of Science ®Google Scholar
• Grason GM. Colloquium: geometry and optimal packing of twisted columns and filaments. Rev Mod Phys. 2015;87:401–419.
   Web of Science ®Google Scholar
• Mandle RJ, Davis EJ, Archbold CT, et al. Apolar bimesogens and the incidence of the twist–bend nematic phase. Chem Eur J. 2015;21:8158–8167.
   PubMed Web of Science ®Google Scholar
• Kang S, Lee E-W, Li T, et al. Two-dimensional skyrmion lattice formation in a nematic liquid crystal consisting of highly bent banana molecules. Angew Chem Int Ed. 2016;55:11552–11556.
   PubMed Web of Science ®Google Scholar
• Ivsic T, Baumeister U, Dokli I, et al. Sensitivity of the NTB phase formation to the, molecular structure of imino-linked dimers. Liq Cryst. 2017;44:93–105.
   Web of Science ®Google Scholar
• Ivsic T, Vinkovic M, Baumeister U, et al. Towards understanding the NTB phase: a combined experimental, computational and spectroscopic study. RSC Adv. 2016;6:5000–5007.
   Web of Science ®Google Scholar
• Lesac A, Baumeister U, Dokli I, et al. Geometric aspects influencing N-NTB transition - implication of intramolecular Torsion. Liqu Cryst. 2018;45:1101–1110.
   Web of Science ®Google Scholar
• Ramou E, Ahmed Z, Welch C, et al. The stabilisation of the Nx phase in mixtures. Soft Matter. 2016;12:888—899.
   PubMed Web of Science ®Google Scholar
• Vanakaras AG, Photinos DJ. A molecular theory of nematic–nematic phase transitions in mesogenic dimmers. Soft Matter. 2016;12:2208—2220.
   PubMed Web of Science ®Google Scholar
• Nishiyama I. Remarkable effect of pre-organization on the self assembly in chiral liquid crystals. Chem Rec. 2009;9:340–355.
   PubMed Web of Science ®Google Scholar
• Efrati E, Irvine WTM. Orientation-dependent handedness and chiral design. Phys Rev X. 2014;4:011003.
   Web of Science ®Google Scholar
• Harris AB, Kamien RD, Lubensky CT. Molecular chirality and chiral parameters. Rev Mod Phys. 1999;71:1745–1757.
   Web of Science ®Google Scholar
• Yoneya M. Toward rational design of complex nanostructured liquid crystals. Chem Rec. 2011;11:66–76.
   PubMed Web of Science ®Google Scholar
• Ocak H, Bilgin-Eran B, Prehm M, et al. Effects of chain branching and chirality on liquid crystalline phases of bent-core molecules: blue phases, de Vries transitions and switching of diastereomeric states. Soft Matter. 2011;7:8266–8280.
   Web of Science ®Google Scholar
• Ocak H, Poppe M, Bilgin-Eran B, et al. Effects of molecular chirality on self-assembly and switching in liquid crystals at the cross-over between rod-like and bent shapes. Soft Matter. 2016;12:7405–7422.
   PubMed Web of Science ®Google Scholar
• Gorecka E, Vaupotic N, Zep A, et al. A twist-bend nematic (NTB) phase of chiral materials. Angew Chem Int Ed. 2015;54:10155–10159.
   PubMed Web of Science ®Google Scholar
• Wöhrle T, Wurzbach I, Kirres J, et al. Discotic liquid crystals. Chem Rev. 2016;116:1139−1241.
   PubMed Web of Science ®Google Scholar
• Fontes E, Heiney PA, De Jeu WH. Liquid-crystalline and helical order in a discotic mesophase. Phys Rev Lett. 1988;61:1202–1205.
   PubMed Web of Science ®Google Scholar
• Roche C, Sun H-J, Prendergast ME, et al. Homochiral columns constructed by chiral self-sorting during, supramolecular helical organization of hat-shaped molecules. J Am Chem Soc. 2014;136:7169–7185.
   PubMed Web of Science ®Google Scholar
• Shu J, Dudenko D, Esmaeili M, et al. Coexistence of helical morphologies in columnar stacks of star-shaped discotic hydrazones. J Am Chem Soc. 2013;135:11075−11086.
   PubMed Web of Science ®Google Scholar
• Ho M-S, Partridge BE, Sun H-J, et al. Screening libraries of semifluorinated arylene bisimides to discover and predict thermodynamically controlled helical crystallization. ACS Comb Sci. 2016;18:723−739.
   PubMed Web of Science ®Google Scholar
• Forman CJ, Fejer SN, Chakrabarti D, et al. Local frustration determines molecular and macroscopic helix structures. J Phys Chem B. 2013;117:7918−7928.
   PubMed Web of Science ®Google Scholar
• Vera F, Serrano JL, Sierra T. Twists in columnar supramolecular assemblies. Chem Soc Rev. 2009;38:781–796.
   PubMed Web of Science ®Google Scholar
• Shcherbina MA, Zeng X, Tadjiev T, et al. Hollow six-stranded helical columns of a helicene. Angew Chem Int Ed. 2009;48:7837–7840.
   PubMed Web of Science ®Google Scholar
• Praefcke K, Eckert A, Blunk D. Core-halogenated, helical chiral triphenylene-based columnar liquid crystals. Liqu Cryst. 1997;22:113–119.
   Web of Science ®Google Scholar
• Lehmann M, Hügel MA. Perfect match: fullerene guests in star-shaped oligophenylenevinylene mesogens. Angew Chem Int Ed. 2015;54:4110–4114.
   PubMed Web of Science ®Google Scholar
• Xu F, Khan IJ, McGuinness K, et al. Self-assembly of left- and right-handed molecular screws. J Am Chem Soc. 2013;135:18762−18765.
   PubMed Web of Science ®Google Scholar
• Diaz-Cabrera S, Dorca Y, Calbo J, et al. Hierarchy of asymmetry at work: chain-dependent helix-to-helix interactions in supramolecular polymers. Chem Eur J. 2018;24:2826–2831.
   PubMed Web of Science ®Google Scholar
• Seddon JM, Templer RH. Polymorphism of lipid-water systems. In: Lipowsky R, Sackmann E, editors. Handbook of biological physics. Vol. 1. Elsevier Science B.V.;  Amsterdam.1995. p. 97–160.
   Google Scholar
• Zeng X, Ungar G, Imperor-Clerc M. A triple-network tricontinuous cubic liquid crystal. Nat Mater. 2005;4:562–567.
   PubMed Web of Science ®Google Scholar
• Saito K, Yamamura Y, Miwa Y, et al. A structural model of the chiral Im3m cubic phase. Phys Chem Chem Phys. 2016;18:3280—3284.
   PubMed Web of Science ®Google Scholar
• Kutsumizu S, Yamada Y, Sugimoto T, et al. Systematic exploitation of thermotropic, bicontinuous cubic phase families from 1,2-bis(aryloyl)hydrazine-based molecules. Phys Chem Chem Phys. 2018;20:7953—7961.
   PubMed Web of Science ®Google Scholar
• Wolska JM, Wilk J, Pociecha D, et al. Optically active cubic liquid crystalline phase made of achiral polycatenar stilbene derivatives. Chem Eur J. 2017;23:6853–6857.
   PubMed Web of Science ®Google Scholar
• Kutsumizu S, Miisako S, Miwa Y, et al. Mirror symmetry breaking by mixing of equimolar amounts of two gyroid phase-forming achiral molecules. Phys Chem Chem Phys. 2016;18:17341—17344.
   PubMed Web of Science ®Google Scholar
• Levelut AM, Germain C, Keller P, et al. Two new mesophases in a chiral compound. J Phys. 1983;44:623–629.
   Web of Science ®Google Scholar
• Pansu B, Nastishin Y, Imperor-Clerc M, et al. New investigation on the tetragonal liquid-crystalline phase or SmQ. Eur Phys J E. 2004;15:225–230.
   PubMed Web of Science ®Google Scholar
• Levelut AM, Hallouin E, Bennemann D, et al. The smectic Q phase, a crystal of twist grain boundaries with smectic order. J Phys II. 1997;7:981–1000.
   Google Scholar
• Vogrin M, Vaupotic N, Wojcik MM, et al. Thermotropic cubic and tetragonal phases made of rod-like molecules. Phys Chemh Chem Phys. 2014;16:16067—16074.
   PubMed Web of Science ®Google Scholar
• Lu H, Zeng X, Ungar G, et al. The solution of the puzzle of smectic-Q: the phase structure and the origin of spontaneous chirality. Angew Chem Int Ed. 2018;57:2835–2840.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Prehm M, Cao Y, et al. Spontaneous mirror-symmetry breaking in isotropic liquid phases of photoisomerizable achiral molecules. Angew Chem Int Ed. 2016;55:312–316.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Poppe S, Dong Q, et al. Isothermal chirality switching in liquid-crystalline azobenzene compounds with non-polarized light angew. Chem Int Ed. 2017;56:10801–10805.
   PubMed Web of Science ®Google Scholar
• Alaasar M, Poppe S, Dong Q, et al. Mirror symmetry breaking in cubic phases and isotropic liquids driven by hydrogen bonding. Chem Commun. 2016;52:13869—13872.
   PubMed Web of Science ®Google Scholar
• Dressel C, Weissflog W, Tschierske C. Spontaneous mirror symmetry breaking in a re-entrant isotropic liquid. Chem Commun. 2015;51:15850—15853.
   PubMed Web of Science ®Google Scholar
• Yan F, Hixson CA, Earl DJ. Self-assembled chiral superstructures composed of rigid achiral molecules and molecular scale chiral induction by dopants. Phys Rev Lett. 2008;101:157801.
   PubMed Web of Science ®Google Scholar
• Perez-Garcia L, Amabilino DB. Spontaneous resolution under supramolecular control. Chem Soc Rev. 2007;36:941–967.
   PubMed Web of Science ®Google Scholar
• Amabilino DB, Kellogg RM. Spontaneous deracemization. Isr J Chem. 2011;51:1034–1040.
   Web of Science ®Google Scholar
• Palmans ARA. Deracemisations under kinetic and thermodynamic control. Mol Syst Des Eng. 2017;2:34–46.
   Web of Science ®Google Scholar
• Roche C, Sun HJ, Leowanawat P, et al. A supramolecular helix that disregards chirality. Nat Chem. 2016;8:80–89.
   PubMed Web of Science ®Google Scholar
• Soai K, Kawasaki T, Matsumoto A. The origins of homochirality examined by using asymmetric autocatalysis. Chem Rec. 2014;14:70–83.
   PubMed Web of Science ®Google Scholar
• Buchs J, Vogel L, Janietz D, et al. Chirality synchronization of hydrogen-bonded complexes of achiral N-heterocycles. Angew Chem Int Ed. 2017;56:280–284.
   PubMed Web of Science ®Google Scholar