Stereochemical rules govern the soft self-assembly of achiral compounds: Understanding the heliconical liquid crystalline phases of bent-core mesogens.

Spontaneous mirror symmetry breaking in achiral fluids is of actual interest for technological applications as well as for the understanding of the spontaneous formation of chirality and complexity in soft matter self-assembly. Here we report a series of achiral bent-shaped 4-cyanoresorcinol bisterephthalates terminated by alkyl chains having a length n ranging from 2 to 22. Some of these achiral compounds with a specific chain length ( n = 12-16) spontaneously form a short-pitch heliconical lamellar liquid crystalline phase. The formation of this helical superstructure is observed at the paraelectric-antiferroelectric transition, but only if it coincides with the transition from random to uniform tilt and with the transition from anticlinic to synclinic tilt correlation of the molecules in adjacent layers. For compounds with a bit longer chains ( n = 18-22) the heliconical phase is only field-induced, but once formed it is stable in a distinct temperature range, even after switching off the field. The presence of the helix changes the phase properties and the switching mechanism in the LC phases from the naturally preferred rotation around the molecular long axis, which reverses the chirality, to a precession on a cone, which retains the chirality once established. These observations are explained by diastereomeric relations between two different and coexisting modes of superstructural chirality, the layer chirality, and the helical twist evolving between the layers.


Polarizing microscopy
The particular textures of the LC phases were recorded by polarization microscopy with a DMRXP, Leica Microsystems. The sample were placed between two microscopy glass slides in a heating stage with temperature controller (Mettler FP82 HT). The textures were recorded with a Nikon Coolpix E 4500 camera or a Leica MC120HD.

DSC
The DSC investigations were carried out on a DSC 7 (Perkin-Elmer) with a constant heating and cooling rate of 10 K/min. The transition temperatures are characterized by the peak temperatures.

Switching experiments and electrooptical studies
For the switching experiments the compounds were filled in commercially available cells (EHC Japan) which consist of two glass plates with a constant distance of usually 6 µm. Afterwards the cell was placed in a heating stage (Mettler FP 82 HT) and the appropriate electric field was applied. The electric field was generated by an AC/DC generator (3322 A, Agilent). The switching response was guided through a resistance cascade (type 1435, FLC Electronics) and tracked by an oscilloscope (TDS 2014, Tectronics).

a) AC field
The AC field experiments were carried out with a triangular wave voltage at a constant frequency of 10 Hz and a resistance of 5 kΩ in PI-coated or non-coated ITO cells 6 µm distance, antiparallel rubbing and a measurement area of 1cm 2 .

b) DC field
The DC field experiments were carried out with a sinusoidal wave at a constant frequency of 20 MHz and a resistance of 5 kΩ in PI-coated or non-coated ITO cells 6 µm distance, antiparallel rubbing and a measurement area of 1cm 2 .

c) Determination of the spontaneous polarization
To determine the spontaneous polarization the area of the switching peak, the cell area and the resistance are needed. The determination also depends on the type of the switching response (antiferroelectric or ferroelectric case).

Phase transitions
A frequency dependence of the imaginary part of the dielectric permittivity for compound 1/12 at selected temperatures is shown in Fig. S20. Three relaxation processes, P1, P2 and P3 were observed in the measured frequency range. The dielectric strength δε and the relaxation frequency f R P2 and P3 are obtained by fitting the relaxation spectra to the Havriliak-Negami equation. The low frequency process P1 is attributed to conductivity. The high frequency relaxation process P3 was observed in the measured frequency range only below 140 o C and S21 is assigned to rotation around the short axis. The plot of temperature dependence of relaxation strength and frequency are given in Fig. S21. Initially, between 140 o C to ~120 o C, the relaxation frequency of P3 decreases and the corresponding relaxation strength increases (by 50%) following a 2S+1 dependence (S being the orientational order parameter), as the sample is cooled down. With further decrease in temperature no significant change is observed in both δε 3 and f R,3 until the sample reaches ~80 °C, after which the δε 3 decreases (from 3.5 to 2.5) and f R,3 increases slightly. The process P2 (medium frequency relaxation process, see Fig.  S20) exists in the measured frequency range in all of the liquid crystal phases. The process can be assigned to the polar switching mechanism and Fig. 10 in the main text shows the temperature dependence of dielectric strength and relaxation frequency for P2.    Figure S28. Chi-scans a) over the small angle range ( = 15-25°) and the wide angle range ( =15-25°) of the 2D XRD pattern of a surface aligned sample of compound 1/16 in the different LC phase ranges; the peak maximum of the small angle scattering are shifted by +1…+2° form 180° and the maxima of the wide angle scatterings are shifted by -2 … -3° from the 90 and 270° positions, meaning that a tilt of 3-5° would be possible, which is within the error (± 3°) of this method.        Figure S37. Development of the spontaneous polarization of 1/18 depending on temperature, measured with the triangular wave method in PI coated ITO cells (1cm 2 , 160 Vpp, 10 Hz).
0 V (SmC s P A ) 0 V (after field treatment, Sm(CP) hel ) -4 V/µm (SmC s P F ) 0 V/µm (Sm(CP) hel ) +4 V/µm (SmC s P F ) Figure S38. Induction of the heliconical Sm(CP) hel phase in the SmC s P A phase at T = 105°C (6 µm, PI-coated ITO cell). Upper row: pristine texture of the SmC s P A phase (left) and texture after application of an AC field (200 V, 10Hz, 2 s, right) and lower line: switching between Sm(CP) hel and the field induced SmC s P F states by rotation on a cone under an DC field.
-2 V/µm 0 V +2 V/µm Figure S39. Switching by rotation around the long axis in the SmA'P F phase of 1/18 at T = 80 °C (6 µm, PI-coated ITO cell) under a DC field. -17° 0° +17° -14° 0° +14° Figure S41. Tilt domains in the field induced planar textures of 1/18 in the SmC s P R [ * ] range at 115 °C (upper row) and in the field-induced SmC s P F phase at 90 °C (lower row) between crossed polarizers (middle) and after rotation of the sample between the crossed polarizers by the given angles (left, right) in a 6 m ITO cell under an voltage of 160 V. S34 120 °C 105 °C 98 °C Figure S47. Inversion of birefringence as observed for a homeotropic sample of 1/20 between crossed polarizers with additional -retarder plate.  Table S1) and rheological properties, and the crystalline phase show a strong hysteresis of the phase transitions (compare DSCs in Figs (Tables 1 and S1) are shown in red. For general abbreviations of the phases, see Table 1; additional abbreviations are explained here: SmA b = biaxial SmA phase; a there is no transition enthalpy for the SmA-SmAP R transition as mistakenly stated in ref. S4; the correct DSC of 1/12 is shown in Fig. 6b; b used as tentative phase abbreviation for SmAP R in ref. S12; c proposed phase structure in ref. S11; d used as tentative phase abbreviation for SmAP R in ref. S10; e the paraelectric SmC s phases were not in all cases designated as SmC s P R or SmC s P R [ * ] ; f SmCP A = SmC a P A ; g field induced phase; h an additional small (0.5 K) range of a SmC s P X phase is observed between SmC S P F hel and SmC s P R at 110-111 °C; for details, see ref. S19; i with surface stabilized SmC a P F state; j SmC s P F is the field-induced structure after removal of the helix; k the precise structure of SmC x P R is still unknown, but it appears to be a polarization randomized, probably heliconical phase.
In a recent report it was shown that at lower temperature the SmC a P A phase of 1/16 is replaced by a non-tilted smectic phase, designated as SmAP A . S23 Herein we propose that this phase should actually be considered as a ferroelectric SmAP F (SmA'P F ) phase. Thus, the situation concerning the tilt in the distinct polar lamellar phases is not simple at all.
Concerning the history of the heliconical phases, the first indication of a heliconical phase structure in bent mesogens actually dates back to 2011 when it was, based on XRD evidences, reported to represent a non-tilted helical organization of the molecules in the uniaxial and polar smectic phase of compound 1/14, and it was therefore named SmAP  ; S10 after recognition of the tilted organization S24 the phase abbreviation was changed to SmCP  S25b and Sm(CP)  S21 and since 2016 SmC s P F hel was used. S5,S19 In 2018 the first report on related heliconical phases with a helix being commensurate with the layer distance, formed by a mesogenic dimer, appeared which was designated as SmC TB . S 26 The designations SmCP  Sm(CP)  for the bent-core molecules were chosen due to the similarity with the incommensurate SmC  * phases of chiral mesogens, S21 and SmC s P F hel indicates that this phase actually represents a structure resulting from the escape of the macroscopic polar order of synclinic tilted polar SmC s P F layers by adopting a heliconical superstructure. S5 Herein we prefer to use Sm(CP) hel as a general phase assignment of heliconical smectic phases, also including possible commensurate phase types.