On the molecular origins of the ferroelectric splay nematic phase

Nematic liquid crystals have been known for more than a century, but it was not until the 60s–70s that, with the development of room temperature nematics, they became widely used in applications. Polar nematic phases have been long-time predicted, but have only been experimentally realized recently. Synthesis of materials with nematic polar ordering at room temperature is certainly challenging and requires a deep understanding of its formation mechanisms, presently lacking. Here, we compare two materials of similar chemical structure and demonstrate that just a subtle change in the molecular structure enables denser packing of the molecules when they exhibit polar order, which shows that reduction of excluded volume is in the origin of the polar nematic phase. Additionally, we propose that molecular dynamics simulations are potent tools for molecular design in order to predict, identify and design materials showing the polar nematic phase and its precursor nematic phases.

To obtain the frequency and amplitude of each relaxation process a simultaneous fitting of the real and imaginary parts of ( ) was performed at each temperature employing the Havriliak-Negami processes 2 : The best fit parameters are shown in Fig. 2 of the main text together with two plots at different temperatures of the derivative of ( ) showing the good agreement of the obtained fits. Although the fit was performed directly on ( ), the reason for using the derivative of the real part in the shown plots is that it allows for much better visualization of the deconvolution of the data into the different relaxations. It is worth mentioning here that the discontinuity at 172 °C in the RM734 data of main manuscript Fig. 2b-c is solely due to the impossibility of uniquely deconvoluting the m ∥, and m ∥, modes above that temperature during the data-fitting process.
The modes heavily overlap in the frequency spectra and resolving them in a robust way without making unsubstantiated assumptions becomes only feasible below that temperature. At higher temperatures we only used one broad mode for the fit (red circles in Fig. 2 between 172 °C and 200 °C) but we imply that both modes are present up to the isotropic phase however we are not able to separately deconvolute them unambiguously.
In the high range of the measured frequencies (1-10 MHz) a third mode is detected, m ∥, . At the I-N transition, its frequency is larger than that of m and rapidly climbs out of the measured frequency range. On further cooling, its characteristic frequency decreases and the mode is again detected. This mode, by frequency and amplitude, can be associated with the rotation around the molecular long axis, as described by the Nordio-Rigatti-Segre theory 3 .
The same reasoning applies to the previously reported RM734 data 4 , which shows the 'splitting' in two modes below 160 °C and a high frequency mode appearing in the measured range at around that same temperature. The anisotropy of the index of refraction n  of RM734-CN was measured by polarization microscopy. A d=20 μm cell with planar alignment (director in the cell plane) was placed between crossed polarizers with the director at 45 degrees with respect to them. The phase difference between the ordinary and the extraordinary light = 2 Δ / was calculated from the intensity of monochromatic light ( = 632.8 nm) transmitted through the sample. Additionally, the temperature dependence of Δ was measured with a Berek compensator in a 9 μm cell. As shown by the figure both results are comparable. Results for RM734-CN are also compared to Δ of RM734 5 (b) Representation of the birefringence vs experimental values of P2 6 . Given the comparable polarizabilities of both materials, the plot reflects the difference in molecular orientational correlations in the N phase of RM734 and RM734-CN.

Supplementary Note 4 -Fredericks transition RM734-CN
A reference value for the splay elastic constant was measured from the change in the dielectric permittivity when a variable voltage is applied to a planar aligned sample. The frequency was set to 30 kHz to avoid undesired ionic effects and the cell's ITO relaxation. Experimental results were fitted to equations: Where the parameter is related to the maximum tilt angle at the centre of the cell ( = sin ( )); the parameters and correspond to the reduced quantities =

Suplementary Note 6 -Conformational distributions calculated from MD simulations
We extracted the conformational distributions of several dihedrals from MD simulations of RM734 and RM734-CN in both polar and apolar configurations at simulation temperatures of 400K as a means to complement torsional potentials calculated at the DFT(M06HF-D3/aug-cc-pVTZ) level which are discussed in the manuscript. Dihedral angles were calculated from the atomic coordinates over the full production MD trajectory and were binned into histograms to give the plots shown below. Differences in conformational populations between the two materials and two configurations are minor.

Supplementary Note 8 -Radial distribution functions
RDFs computed between atoms in the nitro (NO2) or cyano (CN) groups and either the closest carboxylate ester (C(O)O) or terminal methoxy (OCH3) groups. RDFs were computed over the entire production MD trajectory (250 ns total) for RM734 and RM734-CN respectively, in either the polar or apolar configurations. The intermolecular RDF between given sets of atoms is presented as a solid line, whereas the dashed line corresponds to the intramolecular RDF for the same set of atoms.
The RDF presented here are isotropic, being calculated for spherical shells, and are calculated between specific sets of atoms corresponding to functional groups within a given molecule. On the other hand, the pair correlation functions (PCF) presented within the manuscript are anisotropic, being calculated for cylindrical shells oriented with their length along the nematic director, and are computed between the centres-of-mass of all molecules within the simulation.
Representative configurations were selected using selection algebra tools within Pymol; pairs were chosen with a distance between selected group atoms chosen to reproduce a given peak in the RDF. A random pair was picked from a randomly chosen trajectory timestep, the image of the pair was raytraced and the average distance between group atoms was computed.
We link the calculated pairs with the RDF plots using the coloured arrows shown below in Supplementary Fig.  14 and 15.

Supplementary Note 9 -Experimental and calculated scattered intensities for RM734
For each MD simulation, we calculated two-dimensional WAXS patterns as an average of trajectories in the time window 200 -280 ns, as described in the experimental section of the manuscript. For the resulting 2D WAXS patterns, we then calculated orientational order parameters as described elsewhere. Although this is an unconventional way to obtain orientational order parameters from MD simulations, it makes the same assumptions as used for experimental data and so provides directly comparable results. Reassuringly, values obtained from simulated WAXS patterns are not significantly different from the values obtained directly from MD simulations at the same temperature; for RM734 we obtain P2 values of 0.78 and 0.73 in the polar and apolar states, respectively, and for RM734-CN we obtain P2 values of 0.75 and 0.74 in the polar and apolar states, respectively.
Supplementary Fig. 16: Scattered intensity as a function of χ for radially integrated wide-angle X-ray scattering patterns of RM734: (a) experimental data from reference 6 ; simulated data for the polar (b) and apolar (c) nematic configurations, obtained by azimuthal integration of simulated 2D WAXS patterns in the Q range 1 -1.  Fig. 18: RM554. (a) Molecular structures and transition temperatures (°C), (b) Magnetically aligned two dimensional X-ray scattering patterns in the NS phase at 87 °C. X-ray pattern shows the wide angle (100) and low angle (001) scattering peaks characteristic of classical nematic materials and exhibits additional diffuse small angle reflections (002) and (003) similarly to RM734. (c) cylindrical pair correlation function obtained for an MD simulations of RM554 in the polar nematic configuration at 400 K.

Supplementary Note 13 -Solid State of RM734
We revisited the solid-state crystal structure of RM734, which is available from the CCDC as deposition number 1851381. We find that, in agreement with DFT calculations for parallel pairs of RM734 (and also RM554, above), there is a 'close contact' between the carbon atom of the ester (COO) group and oxygen atom of the nitro (NO2) group, with a distance of 3.33 Å (Supplementary Fig. 19). This distance is about 0.1 Å larger than the sum of the VDW radii of the atoms. Fig. 19: Structure of RM734 displayed as a thermal ellipsoid model (50% probability), obtained via X-ray diffraction. The unit cell (space group 1 P ) is indicated. Green lines correspond to nitro-ester close contacts, as described in the text. Viewed perpendicular to the AC plane, along the reciprocal B axis.
We give charges for RM734 and RM734CN in Supplementary Fig. 20, below. We derived charges for other materials studied by molecular dynamics in the same way.