The interplay between spatial and heliconical orientational order in twist-bend nematic materials

: The helical pitch formed by organic molecules, such as the a -helix of proteins, usually requires hydrogen bonding between chiral units and long-range positional order. It was recently found that certain liquid crystal oligomers can have a twist-bend nematic (N TB ) phase with nanoscale heliconical structure without hydrogen bonding, molecular chirality or positional order. To understand the nature of this unique structure, here we present hard and resonant tender X-ray scattering studies of two novel sulfur containing dimer materials. We simultaneously measure the temperature dependences of the helical pitch and the correlation length of both the helical and positional order. In addition to an unexpected strong variation of the pitch with the length of the spacer connecting the monomer units, we find that at the transition to the N TB phase the positional correlation length drops. The helical structure was found not only in the N TB phase but observed even in the upper range of a smectic phase that forms just below the N TB state. The coexistence of smectic layering and the heliconical order indicates a layered (SmA TB ) phase wherein the rigid units of the dimers are tilted with respect to the smectic layer normal in order to accommodate the bent conformation of the dimers and the tilt direction rotates along the heliconical axis. of the heliconical orientational order. We find that below the N-N TB transition the positional order decreases to about 6 nm, while the heliconical-order increases to about 60 nm. These results lead us to propose a refined packing model of the heliconical structure that can explain both the reduction in positional correlations and the temperature dependence of the heliconical pitch. We also show that there approximately 4°C temperature range below the N TB phase where the smectic order and the heliconical order coexist, and we discuss the possibility of a nematic to smectic twist-bend phase transition occurring in the system.


Introduction
The formation of helical structure in molecular systems usually requires crystal or liquid crystal phases with chiral components. The smallest helical pitch formed by organic molecules is the a-helix of proteins 1 with p=0.55 nm, meaning 3.6 amino acids in L-configuration make one turn. 1 Such a tight pitch requires internal hydrogen bonding between chiral amino acid residues that join together in peptide chains that crystallize into a structure with long-range positional order.
The helical pitch of chiral nematic liquid crystals (3-D anisotropic fluids) of rod-shaped molecules ranges from a tenth of a micrometer to several hundred micrometers, i.e., hundreds to thousands of chiral molecules are needed to make one turn. 2 Recently achiral liquid crystal oligomers (dimers [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] , trimers 20,21 and tetramers 12,21,22 ) with rigid arms connected by odd-numbered methylene or ether groups were found to exhibit a twist-bend nematic phase (NTB) 23-25 , which features a heliconical structure with nanoscale pitch, p=6-20 nm range 6,26 . Remarkably such a helical structure requires neither hydrogen bonding, nor molecular chirality or positional order. The application of electrical fields can lead to an unwinding of the helix and the formation of the socalled splay bend phase. 27 Although a number of theoretical models have been developed to describe the heliconical phase structure, important features are still quite poorly understood. Indeed, the very nature of the phase is still controversial. 28,29 Questions range from fundamental issues such as the thermodynamic order of the phase transition to the high temperature nematic phase, the unwinding of the helix with temperature, the nature of the very short pitch of the helix and, crucially, why the rigid extended linear aromatic cores do not nano-segregate from the flexible alkyl chains to form layers with positional ordering instead of building a highly correlated structure with orientational ordering. Models using continuum theory 23, 30 predict that the pitch diverges as p ∝ (TC −T) −1/2 , where the difference between the critical temperature TC and the NTB phase transition temperature TTB (TC-TTB ~ 1°C) is related to the molecular bend and flexoelectric coupling. 31 An alternative theoretical approach is based on microscopic molecular parameters, such as the bend and twist angles (b and a) between the molecular arms, and on pair-pair correlations. 32,33 Small differences in dipole moments and in the linearity of the mesogenic groups of the molecules impact massively the formation and the pitch of the helices, the tilt of the mesogens, and concomitantly elemental parameters of the helical structure such as the radius and contour lengths. For all these questions the correlation with molecular parameters, such as molecular interdigitation and bend angles, are crucial to arrive at a full understanding of the phase.
The first proof of the nanoscale pitch was provided by freeze fracture transmission electron microscopy (FF-TEM) measurements. 6,26 As they require rapid quenching of samples from welldefined different temperatures, detailed temperature dependence measurements would require large number of samples. The temperature dependence of the orientation (bond) ordering using only one sample was first probed with resonant soft X-ray scattering at carbon K-edge (RSoXS) [34][35][36][37] . Although RSoXS can be employed for all materials that contain carbon atoms, the l=4.4 nm wavelength of the soft x-ray limits the resolution to a few nanometers and its l~0.3µm penetration depth requires the preparation of submicron thick films. 38 For dimer materials containing sulfur atom(s) Tender Resonant X-ray Scattering (TReXS) at the sulfur K-edges (E=2.471keV, l~0.5nm, l~10µm) offers a more attractive alternative for precise measurement of the temperature dependence of the helical pitch of the NTB phase, as shown by several recent studies. [39][40][41][42][43][44] In spite of the fact that small angle hard X-ray scattering is not suitable for pitch measurements due to the lack of an electron density modulation coupled to the heliconical structure, recent studies showed that careful analysis of synchrotron SAXS results can provide important information about the molecular associations both in the NTB phase and the N phase above it. 22 Combining SAXS and TReXS measurements allows one to probe the relation between short-range positional and longer range heliconical orientational order in the NTB phase.
In this paper we combine SAXS and TReXS measurements on two novel sulfur-containing analogues of fluorinated dimers with n-pentyl (C5H11) terminal chains. 45,46 In addition to the temperature dependences of the periodicities of the molecular associations and of the heliconical pitch, we also measure correlation lengths of the positional order of the molecular associations and of the heliconical orientational order. We find that below the N-NTB transition the positional order decreases to about 6 nm, while the heliconical-order increases to about 60 nm. These results lead us to propose a refined packing model of the heliconical structure that can explain both the reduction in positional correlations and the temperature dependence of the heliconical pitch. We also show that there approximately 4°C temperature range below the NTB phase where the smectic order and the heliconical order coexist, and we discuss the possibility of a nematic to smectic twistbend phase transition occurring in the system.

A. Materials
The synthesis towards the investigated materials A (Butyl(4''-(7-(2',3'-difluoro-4''-pentyl-       As seen in Figure 3(a), A exhibits two diffuse peaks both in the N (solid lines) and NTB (dashed lines) phases. These peaks are centered at q1 ~ 3.25 nm -1 and q2 ~ 1.53 nm -1 , corresponding to d1 ~ 1.93 nm and d2 ~ 4.10 nm spatial periodicities. As the molecular length is estimated to be 4.0 nm in the fully extended, bent conformation, the presence of two peaks indicates that the separation of the molecules is not only between aromatic and aliphatic groups, but there is also a separation between aliphatic terminal chains and spacers, i.e., both dimer-dimer and monomermonomer interactions occur. The intensity at q1 is about 4 times larger than at q2, representing more populated monomer-monomer (periodicity d1, slightly smaller than half length of the dimer) than dimer-dimer (periodicity d2 corresponding to the dimer length) associations. 22 This ratio is surprisingly different from the parent molecule where only one peak, corresponding to dimerdimer associations, was found in the NTB phase. 46 From the full width at half maxima (FWHM), we estimate the correlation length of the monomer-monomer association to be x=1/FWHM~8-10 nm, corresponding to 2-3 molecular length. For B (Figure 3(b)) with the longer spacer, the peak week, it is only discernible in a semi-logarithmic plot). This additional peak indicates some axially polar dimer-dimer associations within the apolar arrangement of molecules with head-tail symmetry. The peak position of q1 in both A and B is basically independent of the temperature in the smectic phase.
As Figure 4(a) shows, the temperature dependences of the periodicities corresponding to the short-range monomer-monomer associations in the N and NTB phases and of the layer spacing in the smectic phase are similar in the two materials. In the nematic phase, the periodicity of the molecular associations increases on cooling, reaches a plateau about 10°C above the N-NTB transition, and then decreases (especially for A with shorter spacer) before reaching the transition.
Such pretransitional behavior has been observed for a number of dimers [48][49][50] and can be understood as a tilt of the molecular axis in fluctuating NTB domains with respect to the nematic director. 51 The decrease of the periodicity continues below the N-NTB transition, but more weakly in A, and then starts to increase about 8°C above the transition to the smectic phase. In case of B, the period of short-range monomer-monomer associations decreases from 2.19 nm to 2.08 nm, suggesting a tilt of q ≈ cos -1 (2.08/ 2.19) ∼18°. This value is very similar to that found in the non-sulfur containing analogue, DTC5C9, as determined by the ratios of the smallest and largest periodicities of the cybotactic layer spacing measured above and below the N-NTB transition, respectively. 22 However, this apparent tilt is much smaller than that obtained using the ratio of the measured cybotactic layer spacing and the helical contour length between the centers of two mesogens, which yielded q=29° for the DTC analogues. 37 The peak position of q1 in both A and B is basically independent of the temperature in the smectic phase. This is strikingly different from the results for the As seen in Figure 4(b), for both materials the correlation length decreases upon the N-NTB transition from 7.6 nm to 6.6 nm for A and 8.0 nm to 6.3 nm for B, indicating a decrease in the short-range positional order. This is the same trend observed for the DTCnCm materials without sulfur atom. 22 Due to the underlying smectic phase, the correlation length increases gradually in the lower temperature range of the NTB phase, then grows sharply over a ~4°C range (shown by light blue and red highlights in Fig. 4(b)) to over 700 nm and 300 nm in the smectic phase of A and B, respectively.
Our SAXS results provide useful information on the orientational and positional order.
However, they do not give any information about the heliconical pitch in the NTB phase; the latter may be determined from Tender Resonant X-ray Scattering.

D. Tender Resonant X-ray Scattering (TReXS)
TReXS measurements were carried out at beamline 5.3.1 in the Advanced Light Source (ALS), Lawrence Berkeley National Laboratory. A and B were melt-loaded in between two silicon nitride membranes. Birefringence color indicated sample thicknesses in the range of 5-10µm. The samples were attached to a home-made heater which were sealed inside a helium chamber on the beamline. All samples were initially annealed in the isotropic state to remove heat history and defects. The X-ray beam energy was set at the sulfur K-edge by a channel cut double-bounce silicon monochromator. 39 A 2D detector (Pilatus 300K, Dectris, Inc) was used to collect the scattering patterns at a sample-detector distance of 393 mm. The beam center and the sample-todetector distance were calibrated using silver behenate and the smectic A phase of 8CB. All TReXS data presented are measured on cooling at 1°C rate after the samples heated to the isotropic phase. No well-defined features (rings/peaks) are seen in the scattering from the nematic phase.
Intensity vs scattering wavenumber (q) curves for A and B were obtained from the 2D scattering patterns shown in the inset of Figure 5 using the Nika software package. 52 The heliconical pitch (p) was calculated from the peak positions as p=2p/q. The temperature dependence of the pitch for both materials are plotted in Figure 5. This structure can also explain the twisted rope-type texture that appears in the POM images right below the NTB-SmX transition with width and direction matching the stripe texture of the NTB phase (see Figure 2(c) and (g)). These observations strongly indicate that the nanoscale pitch survives in the top range of the smectic phase.
The smectic layer periodicity of ~2 nm is not detected in our TReXS measurements, because the corresponding wavenumber falls outside the range of our detector. After further cooling to the crystal phase, new peaks appear, corresponding to spatial periodicities of approximately 4, 6 and 8 nm, indicating that they are harmonics of the 2 nm periodicity of the monomers.
The temperature dependence of pitch data measured in the NTB phase can be fitted to the expression (1) Here po is the asymptotic pitch value very far from the critical temperature TC, which is slightly (~1°C) larger than the N-NTB phase transition temperature TTB. The parameter Dp is the coefficient of the temperature dependent term, and g is the critical exponent of the temperature dependent term. Since we do not have data up to TC, the four parameters in Eq. 1 appear not sufficiently independent to reliably determine them by least squares fitting; indeed, it is possible to obtain reasonable looking fits with g ranging from 0.2 to 1. For this reason, motivated by predictions of macroscopic mean-field theories 23,30,31 , we elected to fix the value of g to 0.5. Figure 5 corresponding changes in the spatial correlation lengths (see inset of Figure 4(b)) and of the variation of the pitch in the measured range. These indicate that the choice of the exponent g=0.5 predicted by mean-field theory is a reasonably good. However, the fact that po is smaller for the longer molecule B than for A suggests that the exponent g=0.5, expected only to describe the pretransitional temperature dependence of p, does not hold far from the transition. The temperature dependence of the correlation length of the heliconical order for A and B is shown in Figure 6  Inset of (a) shows representative intensity vs q graphs with Gaussian fit to the measured data. Lines are guidance for the eye. Inset of (b) shows the areas below the peaks. Orange and green lines indicate the range where the helix coexist with the smectic order for A and B, respectively.
The temperature dependences of the maximum intensities of the peaks and the area below the peaks are shown in the main pane and in the inset of Figure 6(b). For both materials the intensity is almost constant over the entire range except for a small maximum at 115°C for A and at 124°C for B. Importantly the intensity does not decrease in the last 3-4°C where the smectic phase has already formed, but only within 1°C before the peak disappears. Note, these correlation lengths are an order of magnitude smaller than those estimated for the prototypical cyano-biphenyl type dimer CB7CB from FFTEM measurements. 6,26

Discussion
One of the key observations of our experiments is that on cooling the achiral dimers through the N-NTB phase transition the correlation length of the spatial periodicity decreases, while the heliconical orientational order becomes more correlated. This shows that neither the molecular chirality nor spatial correlation is needed for the formation of the helical structure, and that the heliconical order even suppresses the spatial correlations. Formation of the NTB phase with the NTB phase was first demonstrated. In that material, assuming zero overlap between neighbor molecules, the pitch was found to be related to the cone angle q as (2) where Rmol is the curvature radius of the bent molecules, L is the contour length of the dimer, and k0 is the number of molecules making one full turn of the helix. Applying this equation to our materials, we find that even assuming as large cone angle (q=29°) as was measured for the DTC analogs 37 , the variation of the pitch for B between 125°C and 118°C should be less than 2.6 nm, in contrast to the measured 9 nm (see Figure 5). A similar discrepancy was noticed by increasing length of the spacer due to the flexibility and large number of configurations of the linkage.
As we argued above, the helical structure is due to the coupling between the molecular bend and twist, so the pitch, which is inversely proportional to the twisting power, should decrease as the bend angle decreases on cooling. Such a decrease overcomes the effect of the tilt, explaining the continuous decrease even when the tilt reaches maximum. In fact, the role of the tilt appears to be negligible when the smectic layers form and the cone angle drops to zero. Since the layer spacing was found to be practically constant, the molecular bend angle should be also basically independent of temperature when the layers form. In spite of this, the surviving heliconical pitch slightly decreases, indicating an increasing coupling between the bend and twist on cooling.
Finally, the decrease of the spatial correlation length of the monomer-monomer aggregates below the transition to the NTB phase, and the onset of the smectic layering in the lower part of this phase, indicate a variation of the molecular overlap between two molecules along the helical axis.
Defining an overlap parameter as J=l/L, where l is the length of molecular overlap and L is the contour length of the dimers, for completely zero positional order J randomly varies between 0 and 1 with < J>=1/2, while in the SmA phase with periodicity about half of the dimer length J is uniform and J<1/2.

Conclusion
To summarize, we have measured the nanostructure of two novel sulfur containing dimer materials both by hard Small Angle X Ray Scattering that is sensitive to positional order and by resonant tender X-ray scattering that can detect the heliconical bond order. Our most significant observations are the following: (a) On cooling the dimers through the N-NTB phase transition the correlation length of the spatial periodicity drops, while the heliconical orientational order becomes more correlated. (b) The heliconical pitch is observable even in the upper 3-4°C range of the underlying smectic phase. (c) The temperature dependences of the heliconical pitch show stronger variation near the N-NTB transition than in prototypical CBnCB-type dimers.
We proposed that the coupling between the temperature dependent molecular bend and twist may account for the observed temperature variation and the spacer length dependence of the heliconical pitch.
The observed coexistence of smectic layering and heliconical order -both having periodicities on the scale of the molecular length -is consistent with a SmATB-type phase, where the rigid units of the dimers are tilted with respect to the layer normal to allow for the bent conformation of the dimers, but the tilt direction rotates along the heliconical axis. However, further and more comprehensive experiments are needed to confirm a distinct smectic phase with the twist-bend orientational nanostructure in the dimers investigated here.
Our results demonstrate the value of employing multiple structural probes in order to illuminate the complex interplay between molecular shape, molecular flexibility, and intermolecular packing that governs the microscopic structure of liquid crystalline states, such as the twist-bend phase, that feature novel, nanoscale modulations of the molecular arrangements. The combination of quasi long-range helical orientational order and the strongly varying positional order in these systems represents a particular challenge to structural determination.