On the structure of the Nx phase of symmetric dimers

NMR measurements on a selectively deuteriated liquid crystal dimer CB-C9-CB exhibiting two nematic phases show that the molecules in the lower temperature nematic phase, Nx, experience a chiral environment and are ordered about a unique direction. The results are contrasted with previous reports that proposed a twist-bend spatial variation of the director. A structural model is proposed wherein the molecules show organization into highly correlated assemblies of opposite chirality.

The classical nematic phase (N) with only orientational ordering of the molecules is theoretically and experimentally the LC phase which, due to its fundamental simplicity and technological importance, is by far the most investigated and best understood of all LC phases [1]. Thus the detection of LCs forming two nematic phases is of very high interest, as such observations pose the challenge to test and recalibrate fundamental concepts of LC phase behaviour. Direct transitions between two uniaxial thermotropic nematic phases are typically associated with the formation of more complex molecular aggregates in the low temperature nematic phase, such as column formation in the transition to the columnar nematic phase [2] of discotic or bent core materials [3]. For main chain LCPs , examples have been reported where a first order phase transition can occur between two nematic phases [4].
The recent observation of an additional, low temperature, nematic phase, termed either N x , or N tb , in very simple cyanobiphenyl based dimers with positive dielectric anisotropy, where the mesogens are separated by odd-numbered hydrocarbon spacers [5][6][7][8][9][10][11], and in specifically prepared difluoroterphenyl mesogens [10] with negative dielectric anisotropy, as well as in non-symmetric dimers [12], has sparked a rapidly increasing interest in this class of LCs, and particularly in the structure of this new nematic phase, N x , in thin films and in the bulk. These LCs exhibit characteristic periodic stripe patterns and rope textures in thin films and an electro-optical response typically found in chiral systems, though the molecules are non-chiral [9,10,13]. XRD investigations show clearly the absence of layer reflections, confirmed by extensive miscibility calorimetric studies [5]. The presence of stripes and Bouligand arches with periodicities in the 8-10nm regime in freeze fracture TEM [11] was interpreted as formation of chiral structures on surfaces. Lastly, in a series of papers [7,8,16] focused on an extensive NMR characterization of CB-C7-CB it is argued that the interpretation of the NMR measurements is consistent with a helicoidal conical ("heliconical") nematic phase. Whilst this is in line with a spontaneous twist-bent elastic deformation predicted [14] for the nematic phases of achiral banana-shaped molecules [17], it should be stressed that the structural periodicity of 8-10 nm observed in the Nx phase of dimers [10,11] cannot be readily identified with the periodicity implied by the theoretical proposals of the "twist-bend" nematic phase [14]. Such proposals are based on the continuum theory of elasticity for the nematic phase, involving the elementary (twist, bend, splay) deformations according to the Frank-Oseen formulation of the elastic free energy. However, the validity of the fundamental assumptions of the continuum theory, concerning the slow variations of the director field [1], is problematic when the macroscopic director supposedly undergoes a reorientation of 0 2 over the distance of a half-pitch, which might be as short as one or two molecular lengths. Computer simulations, on the other hand, are not subject to such small length-scale restrictions and there it is found that bent core molecules with flexible arms [15], a model bearing structural analogies with the oddspacer dimers, can form nematic phases consisting of chiral domains which spontaneously self-assemble into larger helical structures.
In this letter we show, based on NMR studies of a selectively deuterated material of the structure CB-C9-CB shown in Figure 1a, that the simple helicoidal conical structure is not the organisation of the N x phase in the bulk. Based on these results, we propose an organisation of the N x phase wherein the molecules form chiral domains or bundles, whose chirality is averaged out macroscopically and whose orientational order defines a unique director.  (1,1,9,9-d 4 ) [18]) and is close to that of non-deuterated CB-C9-CB for which the Nx phase was observed first [5] and to CB-C7-CB [7]. The materials in the N phase are typical, relatively viscous, uniaxial nematics. The onset, through a first order phase transition, of the low temperature N x phase is accompanied by a significant increase of the viscosity (see also ESI-2 [18]).
Figures 1 (b, c) show the quadrupolar spectrum of aligned CB-C9-CB in the temperature range from 110-80 o C. The line shape and the width of the peaks suggest that the sample is uniformly oriented along the magnetic field of the spectrometer.
In terms of 2 H-NMR analysis, the characteristic feature of the N-N x phase transition is the onset of the doubling of the quadrupolar peaks of the CD 2 sites (Figure 1 (b, c)).
Specifically, on entering the N x range, each of the spectral lines of the  -CD 2 groups splits into two lines with splittings 1   and 2   , of essentially equal integrated intensity (see ESI-1.2 [18]) and with continuously increasing separation with decreasing temperature. The quantity is insensitive to temperature, with only a slight increment on cooling, see Figure 2c. While this peakdoubling alone does not allow to directly single-out a particular underlying mechanism, among several possibilities, the mechanism of loss of equivalence of the two deuterated sites [19] as a result of the onset of local chiral asymmetry in the Nx phase is strongly supported by the detailed NMR studies in [7,8] on CB-C7-CB together with other results [20] which suggest the presence of chiral asymmetry.
Accordingly, our analysis and interpretation of the present data is based on adopting the hypothesis of the onset of local chiral asymmetry in the Nx phase on a time scale larger than the time-averaging involved in the NMR measurement. Within this interpretation, the intensities of the two sub-peaks should be strictly equal as they correspond to equal numbers of (orientationally inequivalent) deuterated sites.
The measured splitting in the quadrupolar spectrum of the α-CD 2 group in the nematic phase is related to the respective orientational order parameter Assuming a fixed tetrahedral angle of the a-CD bonds relative to the para-axis of the cyanobiphenyl mesogenic core, we obtain for the order parameter core S of the mesogenic unit the value 0.57; this falls within the typical range for nematics.
The crucial NMR experiment for testing the formation of the helicoidal conical structure in the bulk makes use of the high viscosity of the Nx phase. This allows collection of NMR spectra with the sample oriented parallel as well as perpendicular to the magnetic field. Specifically, when rotating the initially aligned sample to 90˚ with respect to the magnetic field, there is sufficient time to measure the NMR spectrum in this configuration before the magnetic field reorients the sample. Figure   2a shows the quadrupolar spectrum at 90 ˚C before (red) and after (blue) a rotation of the sample by 90˚ with respect to the magnetic field. Notably, the spectrum of the  configuration of the nematic director for a heliconical structure is given in Figure 3a.
In this model the director follows a configuration described by  Clearly, there is no helical, nor heliconical, molecular ordering within the chiral domains. Instead, the dimer molecules within these domains assume predominantly twisted, chiral, conformations [21] and the enantio-selective local molecular packing confers to the domain its chirality. As the molecules are statistically achiral, each to the isotropic phase and then slowly cooled to the desired temperature. The splittings obtained from different cooling runs agree within an error of less than 1%.

Analysis of measured spectra.
(a) Aligned sample spectra.
The doubling of each of the quadrupolar peaks of the α-CD 2 sites in the N x phase produces two sub-peaks of different heights and widths. Specifically, the height of the outer sub-peak is appreciably lower but its width is larger than that of the inner subpeak. The integrated intensity, as it comes out from the direct manual integration in the NMR software is about 10% smaller for the outer peaks. However, this neglects the fact that a part of the outer peaks (the asymmetric 'foot' to the middle) lies

(c) Simulation of the aligned and 90 o -flipped spectra in the N X phase.
In figure S2 we present the experimental NMR spectra of the aligned (red) and of the flipped by 90 o with respect to magnetic field samples, (see also fig Fig 2(a) Here, , 0 H n θ = or 90, denotes the angle between the magnetic field and the director for the aligned and the 90 o -flipped samples, respectively.
as expected for a uniaxial nematic. Figure S2: 2 H NMR spectra of the aligned sample (red) and after a flip by 90˚ with respect to the magnetic field (blue) in the Nx phase at 90 o C. The thick lines (magenta for the aligned sample and cyan for the 90˚-flipped) are simulated spectra assuming Lorentzian line-shapes according to eq Error! Reference source not found., with the best-fit parameter values listed in Table I.

The effects of viscosity on the 90 o -rotated and spinning sample spectra.
At 80°C, spectra recorded under continuous spinning (Figure 2(c)) showed no significant difference when recorded with spinning rates of 6 rpm, 60rpm, 100 rpm and 200 rpm. This independence on the spinning rate indicates that a uniform cylindrical distribution of the nematic director is generated by the spinning of the sample and that such distribution is not influenced by hydrodynamic effects, as these would show a dependence on the spinning-rate. Furthermore, the spectra had not relaxed back to equilibrium two hours after the rotation was stopped. Such slow relaxation rates, together with the absence of significant hydrodynamic effects on spinning, are direct implications of the high viscosity of the N X phase.
The orientational relaxation times show a relatively rapid variation with temperature within the N X phase and a dramatic drop on entering the N phase. Thus, the spectra shown in Figure 2

Further difficulties with the heliconical interpretation.
As detailed in the main text the primary inconsistencies regarding the interpretation of the experimental data with the heliconical model of the director configuration are (i) the measured 90 o -rotated and spinning sample NMR spectra indicate rather directly that the entire sample exhibits a single, uniform, director, and (ii) the structural periodicity of 10nm sets a rather short length scale on which not only the description of a continuous twist and/or bend spatial variation of the director would be invalid, but even the validity of the definition of a nematic director would be questionable.
Notably, chiral domains in the nematic phase of achiral compounds exhibiting dimerisation through hydrogen bonding were reported more that 15 years ago 1 ; however, such domains were found to exhibit macroscopic twist deformation of the director in the optical regime.
In addition to the above deficiencies, the heliconcal interpretation is in difficulty with the consistent reproduction of the orientational ordering implied by the NMR measurements and also with the slow orientational relaxation observed directly in the N X phase, as described in section 2. These additional difficulties are discussed separately below.

Spectral line-shape for a hypothetical heliconical configuration with the magnetic field perpendicular to the helix axis
Here we present the calculation of the NMR spectrum of a sample assuming that the N X phase is a twist-bent nematic presenting a heliconical distribution of the director field. The calculated spectra are then contrasted with the measured spectra.
As indicated in the main text of the paper, the splitting associated with an α-C-D bond, is given by the ensemble average: In eq. (S.7) the parameters  It is clear from equations S.5 to 7 that the measured NMR spectrum can pick up variations of the angle of the director relative to the spectrometer magnetic field and that these variations are manifested as broadening of the spectral lines, in excess to their width at perfect alignment. The resolution of the experimental method sets a lower bound to the extent of variations that can be detected unambiguously on the measured spectra. With the simulation procedure described above we have found that, within the resolution of the experimental spectra, both the aligned sample spectrum and the 90 0 -flipped spectrum place an upper bound of 5 0 for possible variations of the angle of the director relative to the magnetic field. Accordingly, if it is assumed that, as a result of a hypothetical heliconical distribution, the director forms a constant angle θ 0 with the magnetic field, then it is directly deduced from the 90 0 -flipped spectrum that this angle should be below 5 0 . Of course variations of such a limited range could also be due to minor fluctuations of the director within the sample, not related to any heliconical distribution. In any case, it should be noted that the upper bound for θ 0 <5 0 , set by the combined experimental and simulation resolution, is far below the values given for θ 0 in the literature 3 on the twist-bend nematic phase.

Order parameters of the mesogenic units
The relatively low values implied by the NMR measurements for the order parameter . Accordingly, the mechanism of fast molecular diffusion could be applicable in the case of the N phase rather than the N X.