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First-Principles Experimental Demonstration of Ferroelectricity in a Thermotropic Nematic Liquid Crystal: Spontaneous Polar Domains and Striking Electro-Optics

We report the experimental determination of the structure and response to applied electric field of the lower-temperature nematic phase of the previously reported calamitic compound 4-[(4-nitrophenoxy)carbonyl]phenyl2,4-dimethoxybenzoate (RM734). We exploit its electro-optics to visualize the appearance, in the absence of applied field, of a permanent electric polarization density, manifested as a spontaneously broken symmetry in distinct domains of opposite polar orientation. Polarization reversal is mediated by field-induced domain wall movement, making this phase ferroelectric, a 3D uniaxial nematic having a spontaneous, reorientable, polarization locally parallel to the director. This polarization density saturates at a low temperature value of ~ 6 microcoulombs/cm-sqd, the largest ever measured for an organic material or for any fluid. This polarization is comparable to that of solid state ferroelectrics, and is close to the average value obtained by assuming perfect, polar alignment of molecular long axes in the nematic. We find a host of spectacular optical and hydrodynamic effects driven by ultra-low applied field (E~1V/cm), produced by the coupling of the large polarization to nematic birefringence and flow. Electrostatic self-interaction of the polarization charge renders the transition from the nematic phase mean-field-like and weakly first-order, and controls the director field structure of the ferroelectric phase. Atomistic molecular dynamics simulation reveals short-range polar molecular interactions that favor ferroelectric ordering, including a tendency for head-to-tail association into polar, chain-like assemblies having polar lateral correlations. These results indicate a significant potential for transformative new nematic science and technology based on the enhanced understanding, development, and exploitation of molecular electrostatic interaction.


Introduction
The first theoretical treatments of collective molecular orientation in liquids, by Debye [1] and Born [2], were electrostatic versions of the Langevin-Weiss model of the paramagnetic/ferromagnetic transition in solids [3]. Born envisioned the orientational ordering of rod-shaped molecules of a nematic as a phase transition, the proposed ordering mechanism being the interaction of molecular electric dipoles, so that the resulting nematic phase was ferroelectric, i.e., predicted to have a spontaneous non-zero polarization density. Thus, the notion of LCs with polar order, introduced more than a century ago, has grown as a field of broad interest and challenge at the frontiers of LC science, stimulating rich themes of novel chemistry and physics [4,5,6 7,8,9,10,11,12,13] However, following Born's model, some calamitic molecules without molecular dipoles were found to exhibit nematic phases [14], while ferroelectricity failed to materialize as a molecular nematic phenomenon. Born's calculation thus appeared to be incomplete, stimulating a variety of different models of nematic ordering in which both steric and/or electrostatic interactions were considered. These included the Maier-Saupe theory [15], where steric interactions produced apolar (quadrupolar) order, and others in which the nematic ordering could also be polar [4,5,16,17,18,19,20,21,22]. The appearance of polar ordering in these models and Born's is considered to be an equilibrium transition between bulk phases of higher and lower symmetry [23,24]. The models propose order parameters constructed to characterize this change of symmetry, and develop benchmarks for experimental testing, predicting pretransitional behavior as the phase transition is approached from higher or lower temperature, as well as describing the properties of the polar ordered phase and its distinct symmetry-related states. In the case of relevance here, of a uniaxial, non-polar nematic transitioning to a uniaxial, polar nematic with the polarization along the director, there are two ordered states related by reflection through a plane normal to the polarization. If such states coexist in a sample, they must form reflection-related domains with opposite polarization separated by well-defined domain walls [25,26,17]. Polar domains and their boundaries are also described by the models, specifically by the elasticity and order parameter energetics of the polar phase, making the domains the signature features of spontaneous polar ordering to be probed and understood in characterizing the nature of the phase transition. If such domain boundaries can be moved or removed by application of a field then the mean polarization can be changed, and if this motion is irreversible, then the polar phase will exhibit switching and hysteresis as emergent properties and can be considered macroscopically ferroelectric [27]. Here we present the direct observation of such spontaaneously broken symmetry in the form of domains of opposite polarization, grown without applied electric field, as a first-principles demonstration of ferroelectricity in a thermotropic, uniaxial, nematic liquid crystal of rod-shaped molecules.
In 2017 Mandle, et al. [28], and Kikuchi et al. [29] separately reported new liquid crystal (LC) compounds exhibiting unusual phase behavior: two distinct, fluid nematic phases separated in temperature by a weakly first-order phase transition. In both cases, the molecules were rod-shaped, with several intramolecular dipoles distributed along their length whose projections onto the molecular long axis summed to a large overall axial dipole moment of ~ 10 Debye. The high temperature phase of both ! ! 4 mesogens was reported to be a typical nematic, whereas the low temperature phase exhibited enhanced dipolar molecular associations, reported to be antiparallel in the Mandle system [28], and suggested to be parallel in the Kikuchi system, the latter being termed "ferroelectric-like" [29], giving macroscopic polar ordering in response to an applied electric field. Mandle et al. subsequently synthesized a number of homologs of their molecule in an effort to develop structure-property relationships for this phase [30], and pursued, in collaboration with the Ljubljana group, a series of physical studies on one of these (RM734), shown in Fig. 1A, leading to the claim that this phase was locally polar but, on some longer scale, an antiferroelecric "splay nematic" [31,32,33,34,35], a splay-bend phase of the type originally proposed by Dozov [36].
Our resynthesis of RM734 (SI Appendix Sec. S1) and observation of its electro-optic behavior using polarized light microscopy provides no evidence for a splay nematic phase, but rather leads us to the unambiguous conclusion that, upon cooling from the higher temperature, non-polar, uniaxial nematic (N) phase, RM734 undergoes a transition to another uniaxial nematic (NF) phase that is ferroelectric. The key evidence for this result is the first observation in a nematic liquid crystal of the defining characteristics of ferroelectricity: (i) the formation, in the absence of applied electric field, of spontaneously polar domains of opposite sign of polarization separated by distinct domain boundaries; and (ii) field-induced polarization reversal mediated by movement of these domain boundaries. These phenomena are summarized in In the N phase, the local texture of the planar-aligned cell described in Fig. 1 is optically featureless. On cooling toward the NF phase, a random pattern of stripes extended along the buffing direction appears, but once in the NF phase these stripes coarsen, leading to a texture that is again local optically featureless (Figs. 1B-D) but characterized on a larger scale by a pattern of well-defined lines, some delineating distinct, lens-shaped domains 100 µm or more in extent (Figs. 1D-G), all formed in the absence of applied electric field. Application of an ultra-small (~1 V/cm), in-plane, DC test field, E, applied parallel to the in-plane buffing and director n(r), shows that for E > 0 the director inside these domains begins to reorient while the orientation outside remains fixed (Fig. 1E), whereas for E < 0 the region outside the lens-shaped domains reorients and the orientation inside remains fixed (Fig. 1G), indicating that the domain boundaries separate regions with opposite response to in-plane field, and therefore of opposite inplane polarization. Increasing the field causes the domain boundaries to unpin, shrink, and disappear ( Fig. 1F), moving hysteretically to increase the area of the field-preferred state. As E is increased, if the polarization P(r) in the preferred state is not perfectly parallel to E, then it must become increasingly so in order to reduce P!E. The optical characteristics of this field-induced condition will then reveal the resulting structure of n(r). We find that with E applied, n(r) becomes more parallel to E, i.e., E, P and n become co-linear (E parallel to P and n either parallel or antiparallel to P). However, this field-induced condition of the preferred state differs only slightly from its initial field-free condition stabilized only by the buffing, which couples to n(r) (for example, the outside of the lenses in Fig. 1E and the inside of the lens in Fig.   1G). This can be the case only if n(r) is either parallel or antiparallel to P in the field-free NF state. The observations of Fig. 1, constituting a first-principles demonstration of nematic ferroelectricity, are described in more detail below and in SI Appendix Secs. S3,S4.
gestive of a relatively weak coupling to the azimuthal anisotropy of the surface. Measurements give a uniaxial birefringence !n ~ 0.2 (SI Appendix Fig. S17), suggesting that the alignment is planar, with n(r) nearly parallel to the plane of the plates. Tilting the cell away from being normal to the light beam did not reveal significant tilt of n(r) out of the cell plane, but a small (few-degree) pretilt may be present. In addition to the locally uniform, planar texture imposed by the surfaces upon cooling into the N phase, we observed a few twisted areas (SI Appendix Fig. S6A), but generally the local preferred orientation was the same on the two plates and therefore likely established by a combination of the buffing with surface memory [38] of the nematic director pattern as it was first growing out of the isotropic.
Ferroelectric Nematic Phase -Upon cooling through the N -NF transition, the cell becomes patterned with an texture of irregular domains extended locally parallel to n(r), first appearing on a submicron scale but then annealing over a roughly 2ºC interval into a pattern of elongated lines of low optical contrast that are and also oriented generally along n(r). Some lines coarsen to form closed loops, 10-200 microns in extent, having a distinctive characteristic lens shape, elongated along n(r). Sample textures from this evolution are shown in Fig. 1, with additional images in SI Appendix Figs. S4-S9. Upon completion of the transition, apart from these loops, the texture is smooth and very similar to that of the N phase at high T (SI Appendix Fig. S6). A small increase of !n is observed at the transition (SI Appendix, Fig. S13), and !n continuously increases with decreasing temperature, T, behavior in agreement with that previously reported for the nematic order parameter [32].
The NF phase exhibits striking electro-optic behavior in DTLM, starting with the response to the application of tiny (< 1 V/cm) in-plane electric fields. This sensitivity was exploited to probe and understand the static and dynamic changes of n(r) and P(r) in applied electric fields. Typical textures observed in these experiments and the responses used to identify ferroelectric domains and determine their polarization orientation are indicated in Fig. 2. In the absence of applied field, the LC director n in these cells is ! ! 6 generally along the buffing direction z and we observe domains separated by distinct boundaries. As in Fig. 1, one set of domains responds only weakly to fields applied along z, indicating that in these regions P is already parallel to E, and therefore that before the field is applied P is everywhere either nearly parallel to or antiparallel to z. In an applied field, in the domains where P is nearly antiparallel to E, the polarization responds by rotating toward E. The optical response to test fields makes the difference in polarity readily distinguishable, leaving no doubt that these domains are polar. The green vectors in Fig. 2 indicate the field-induced reorientation of P(r) in the mid-plane of the cell: n(r) at the cell surfaces in this experiment remains parallel to z.
In an applied electric field, the n(r) field of the preferred states becomes slightly better aligned along the field direction, evidence that n and P are parallel. We considered the possibility that the NF phase is biaxial, with n along z and secondary directors oriented preferentially parallel and normal to the cell surfaces. When such a biaxial phase grows in from a higher temperature uniaxial phase, the cell generally exhibits arrays of characteristic disclination lines where 180º flips of the secondary directors have become trapped at one or both surfaces [39]. If such structures were present in our cells, they would show up clearly in the polarized light microscope, but none are observed. We conclude that the bulk NF condition is uniaxial, with n parallel to P in the absence of applied field. When an electric field is applied, n(r) experiences electric torques through its coupling with P, which we assume remains substantially locally parallel to n, i.e., that the local optic axis is parallel to P. Fig. S4 presents a larger area of the t = 11 µm cell in orientations having the average director along the crossed polarizer/analyzer direction (A, F-H), and at 45º to it (B-E), illustrating that the overall textural alignment gives reasonably good extinction between crossed polarizers, but with some brighter regions allowed because of the softness of the anchoring as noted above. The images show polarization reversal driven by an adjustable DC in-plane electric field. The textures of n(r) in the limiting states of plus or minus E are identically black, but separated by a striking scenario of domain wall formation, coarsening and disappearance, all in the tiny DC field range -2 V/cm < E < 2 V/cm. The field-aligned states extinguish between crossed polarizers as in (A), meaning that that they have n(r) everywhere parallel to z, and show a pink birefringence color in the third-order Michel-Levy band as in (B). The intermediate states have n(r) in the y,z plane but with spatial variation of its azimuthal orientation !(r) about x. This lowers the effective retardance of these regions, moving their birefringence down into the second-and first-order Michel-Levy bands and producing intense birefringence colors. The uniformly oriented domains obtained following field reversal are states in which the n,P couple has been reoriented in the bulk LC and also flipped on the aligning surfaces, the latter mediated by domain wall motion.
The field-induced reorientations in the planar-aligned geometry of Figs. 1,2 are twist deformations of the azimuthal orientation !(r) of n(r) about x, having the form !(r) = !c(x) Cos[✂x/t] for small"!c, where !c(x) is the reorientation pattern in the cell midplane. This deformation can be generated in a uniaxial dielectric N phase using an in-plane AC electric field to induce a twist Fréedericksz transition, for which the threshold field will be given by E! = (✂/t)!(K/#o!#) [40]. Assuming a cell gap t = 11 µm, and typical nematic values of Frank elastic constant K ~ 5 pN, and a dielectric anisotropy !# ~ 5, one ! ! 7 finds E! ~ 1000V/cm, giving an estimate which sets the field scale for typical in-plane dielectric nematic electro-optics. The fields required to produce the reorientations in the NF phase in Figs. 1,2 are three orders of magnitude smaller. The polar coupling to field indicates that our observed field-induced reorientations are polar Fréedericksz transitions with P(x) starting antiparallel to E, for which the torque balance equation is KT!xx + PEsin!(x) = 0 [41,40]. This gives a field threshold of EP = (✂/t) 2 (KT/P), and a cell midplane reorientation of !"$%&"'"![6(E -EP)/EP], so that a !" = 0 to !" = 90º reorientation occurs in the field range EP < E0-90 < 1.4 EP. Measurements of E0-90 yield an experimental value of the threshold of EP ~ 1 V/cm, from which we can estimate P. Taking KT to be in the range 2 pN < KT < 5 pN gives a value for P in the range 3 µC/cm 2 < P < 6 µC/cm 2. .
We also measured P(T) directly from the field-induced current associated with polarization reversal. We used both square-and triangle-wave driving fields in several different, two-terminal cell geometries, including: an in-plane cell similar to that used for the electro-optics but with gold electrodes (Fig. S3, SI Supplement); 100 µm-thick sandwich cells with conventional gold electrodes; and a 0.5 mm diameter, cylindrical capillary with the LC in a 150 µm gap between planar electrode faces oriented normal to the cylinder axis (Fig. S3, SI Supplement). Time integration of the current signals obtained using these geometries produced consistent values of the polarization density as a function of temperature. Typical polarization data obtained from the in-plane cell using a square-wave drive frequency f = 200 Hz are shown in Fig. 3. The cell current showed only small capacitive and resistive contributions in the N phase, but, upon cooling into the NF phase, became dominated by an additional peak which exhibited the characteristics of ferroelectric LC polarization current: carrying a driving amplitude-independent net charge Q (from which P was calculated); and exhibiting a risetime ( varying inversely as the drive amplitude (Fig. 3C).
The resulting P increases continuously from small values at the transition, saturating at low T at a value P ~ 6 µC/cm 2 , a value comparable to the ~4 µC/cm 2 found by Kikuchi et al. in the compound DIO [29].
Polarization in the N -NF transition temperature region may include pretransitional contributions in the N phase, but this has not yet been studied in detail.
The estimate above of P from the polar twist threshold EP at T = 120ºC agrees well with the P data in Fig. 3: the magnitude of spontaneous polarization achieved field-free in domains grown and cooled from the N phase is comparable to that induced by a field in the NF phase, as expected for a ferroelectric.
Further significance of P ~ 6 µC/cm 2 can be appreciated by calculating a polarization estimate Pe Debye is the axial molecular dipole moment of RM734 [30] (SI Appendix Sec. S10) and v is the volume/molecule in the phase, v = 325 cm 3 /mole = 540 Å 3 /molecule, assuming a LC mass density of ) = 1.3 g/cm 3 (SI Appendix Sec. S6). Using these parameter values and assuming complete polar ordering of the molecular long axes, we find Pe ~ 6.8 µC/cm 2 , comparable to our measured P at low T and indicating that RM734 has extremely strong spontaneous macroscopic polar ordering, a condition impossible in any state with domains of competing polarization as in the proposed splay nematic, for example. This value of P is confirmed by our atomistic MD simulations, which yield P " 6.2 µC/cm 2 in an equilibrated simulation of 384 molecules polar ordered in the NF phase, as discussed below and in SI Appendix Sec.

S10.
This P values is roughly six times larger than the highest polarization ever achieved in tilted calami-! ! 8 tic or bent-core chiral smectic LCs [42,54]; is comparable to that found in polar columnar phases [43]; and is well within the range exhibited by solid-state ferroelectrics [44]. This result, combined with our textural observations, indicate that the NF phase is a 3D fluid, macroscopically homogeneous, polar uniaxial nematic phase. The agreement with the spontaneous P measured from the polar twist threshold indicates that this is the ferroelectric state.
Given these very large polarization values, it is useful to summarize several of the relevant features of polar electro-optic, electrostatic, and elastic behavior, developed in the study of chiral smectic ferroelectric LCs, which can now be expected for the NF: (i) Polar Twist Fréedericksz transition [40,41] -We take the uniform equilibrium state to have P along z. In small applied fields, electrical torque on the director field *E = P x E comes from the coupling of field to polarization. Applying this coupling in the description of the twist Fréedericksz transition, the equation describing the azimuthal orientation field !(x) across the thickness of an otherwise uniform cell becomes KT!xx + PEsin! = 0, with the threshold characteristics given above. (ii) Boundary penetration -Solving K!(z) zz + PEsin!(z) = 0 instead in the y,z plane, with the boundary condition !(z=0) = 0, using the one-elastic constant (K) approximation, and applying an electric field to stabilize !(z) = 180º at large z, a # reorientation wall is established in the LC near z = 0, given by !(z) = 4tan -1 [1 -exp(-z/+E)]. The ferroelectric field penetration length +E = !K/PE gives the approximate width of the wall [45], the distance that the effects of local orientational pinning such as the wall can penetrate into the neighboring LC, assuming the latter has P held in place by E. The penetration depth is +F ~ 1 µm for an applied field E = 1 V"#"cm). (iii) Block polarization reorientation and expulsion of splay (splay-elastic stiffening) [46,47,48,49,50,51,52,53] -Spatial variation of P(r) generally results in bulk and surface polarization charge density, given respectively by )P = !!P(r) and $P = P s "s. The electric field generated by the bulk charge opposes the bulk distortion of P(r) that caused it, producing a bulk energy UP = $%dv !!P(r)!!P(r')[1" #" |(rr')|]. Assuming a periodic transverse modulation "Py(r) of amplitude P"ny and wavevector qy, so that !!P(r) = ✦ Py(y)y = iqyPz"ny in our geometry, we have an elastic energy density Usp = $[K s qy 2 + 4#P 2 "#"#]#|"ny| 2 , meaning that the polarization term will be dominant for qy < #!,"#"+P, where +P = !#K/P 2 is the polarization self-penetration length. Since for P = 6 µC/cm 2 we have +P ~ 0.1 nm, this dominance will act down to molecular length scales. The result is that low-energy elastic distortions of the n,P couple allow only bend, with splay of n(r) and P(r) expelled from the bulk and confined to reorientation walls of characteristic width +P. On the other hand, if we consider a longitudinal modulation "Pz(y,z), the additional electrostatic free energy density will be UP = $[ 4#P 2 qz 2 #" #(qz 2 + qy 2 )]# |"Pz| 2 [48]. All such polarization-based effects are reduced by free space charge, such as ionic impurities in the LC and its containing surfaces [49,50], ionization of the LC itself, and charge injected from the electrodes, all of which tend to screen the polarization charge. In SmC* FLC cells, when the polarization is small (P < 20 nC/cm 2 ) the bound polarization charge can be substantially screened but for large polarizations (P > 100 nC/cm 2 ) the free charge supply can be exhausted and polarization effects manifested. For the largest SmC* polarizations (P ~ 800 nC/cm 2 ), the polarization charge is ! ! 9 largely unscreened and the polarization effects are quite dramatic [54]. (iv) Field-step reorientational response -The reorientation dynamics leading to polarization change and the electro-optic (EO) response to changes in applied field are complex, depending on elastic, viscous, surface, and flow-induced torques in addition to that of the field. However, with the application of a large field step the electrical torques initially dominate and these determine the risetime of the optical response. The balance of field and viscous torques gives a characteristic reorientation risetime on the order of ( = !1"#$%%"&'()("!1 is the nematic rotational viscosity [45,55] (Fig. 3C).
The electro-optic response of the NF phase in a cell with in-plane electric field applied shows uniquely polar features, with P(r) reorienting in the y,z plane through the azimuthal angle !(r) about x, determined by the local surface, elastic and electric torques. Buffed surfaces stabilize two planar-aligned states (!=0 and !=#) with opposite signs of P(r), so the cell has four stable states, two that are uniform and two that are twisted, illustrated in Fig. 4. These equilibrium states are separated by # surface disclination walls (magenta dots in the section drawing of Fig. 4). If complete polarization reversal is to be achieved by an applied field, P must be switched on both surfaces. We refer to domain boundaries such as these, where both n and P reorient but maintain a fixed relative sign, as Polarization-director (P-n) Disclinations (PnDs). In a texture with only PnDs the local orientation of P(r) relative to n(r) will be either parallel or antiparallel everywhere. In Fig. 4, an applied field preferring the uniform (U) dark state is deforming a central domain that had polarization that was initially opposed to the field and is now partially rotated toward it (green vectors). This central domain result is a twisted-untwisted (TU) state in which the director twists along x from the surface-preferred alignment parallel to z at one cell plate through azimuthal angle !(x) to the field-aligned orientation in the midplane of the cell, and then twists back to the surface-preferred alignment on the other glass plate. The field causes that domain to shrink, moving and eventually eliminating the disclination walls in order to achieve complete polarization reversal. The motion of the walls on the two cell surfaces is different because the pinning strengths are different and spatially inhomogeneous, with the result that they do not remain in register, leading to the formation of the olive (left hand, TL) and gold (right hand, TR) twisted states seen surrounding the central domain, and where the central TU domain color (green, blue, or pink) depends on the degree of fieldinduced reorientation of n,P in the sample mid-plane. Fig. S4 show several other modes of field-induced polarization reversal.

Fig. 5 and SI Appendix
The initial response of a uniformly aligned region to an increasing in-plane DC field in the range 0 < E < 2 V/cm that opposes the local polarization is to form a zig-zag modulation in the orientation of n(r) and P(r), illustrated in Fig. S4, in which the non-zero spatial variation is -n(r)/-z, along the director, making it a bend wave. Bend has a lower polarization space charge energy cost than a splay wave (nonzero -n(r)/-x), which would generate stripes parallel to n rather than normal to it. As the field strength is increased, the degree of reorientation increases and distinct boundaries appear between the half-periods of the modulation, separating stripes of uniform internal orientation. Fields of a few V/cm drive complete (+✂, -✂, +✂, -✂) reorientation of n in the cell midplane for which these boundaries become 2# walls, sub-optical resolution (~+P) in width. This process is even more dramatic with dynamic driving as illus-! ! 10 trated in Fig. 5A which presents snapshots of alternating field reversals generated by a 5 Hz AC triangle wave field of peak amplitude up to Ep = 10 V/cm. As the field is increased from zero, the stripes become very regularly and narrowly spaced. The zig-zag pattern of the stripes indicates an overall structure where Pz is constant, ensuring that there is no net polarization charge at the stripe boundaries, and where the backflow induced in each stripe matches that of its neighbor. The field is not strong enough to reverse the surfaces in this case. In Fig. 5B polygonal domains are formed upon reversal in which chargestabilized areas of uniform P are bounded by sharp domain boundary lines, each oriented along a vector l such that P!l has the same value on either side of the line, as in panel 4, where the angular jump in n(r) is 90º. This geometry reduces the net polarization charge on the line. Similar structures are found in high-P chiral smectic ferroelectrics [56] and in ferromagnets [57]. The textures of the charge-stabilized domains can also be employed to visualize directly the reorientation of P(r) under applied field, as shown in Fig.   5C, where a circular air bubble enables tracking the polar orientation of n(r) during field reversal. The n(r),P(r) structures are sketched below each panel. The director is anchored tangent to the bubble surface, resulting in a director field that is largely bent around the bubble, with splay concentrated in two 180º wedge disclinations (red dots). The line connecting these defects is parallel to n(r) in the area surrounding the bubble.

Nematic Ferroelectrohydrodynamics -
The polarization density of the NF phase creates a fluid which is extraordinarily responsive to both external applied fields and its internally generated polarization space charge. While the discussion above has focused on the effects of field-induced molecular reorientation, the most interesting and useful effects of the NF may be its ferroelectrohydrodynamic or ferroelectrorheological behavior, exemplified by the observations shown in Fig. 6 and SI Appendix Sec. S7. In this experiment, RM734 is filled into a t = 10 µm cell with random-planar alignment of n. An in-plane electric field is ternately through +# and -# domains, giving the radial texture in Fig. 6A. Thus, dynamically P(r,t) is everywhere parallel to E(r,t) and v(r,t) when voltage is applied. The fact that the product P(r)!E(r) is unchanged by applied field reversal and yet v(r) changes sign, indicates that )(r) does not change sign with P(r), i.e., the driving has caused the fluid to become charged. SI Appendix Fig. S16 shows the structure of P-reversal bands in the neighborhood of the electrode gap.
We measured vi, the initial value of the defect velocity upon the field reversal at the location indicated in Fig. 6A. This velocity depends dramatically on temperature, as shown in Fig. 6C, with flow being essentially absent in the N phase and commencing upon cooling through the N-NF transition. The veloci-! ! 11 ty eventually decreases with decreasing T, presumably because of the increasing viscosity of the LC.
Thus, an applied electric field promotes the creation of regions with positive charge density.
Charging of the NF by AC field application is to be expected due to its polar asymmetry. Electrode surfaces contact NF material having the sign of P alternating in time. The NF, because of its polar symmetry, has diode-like polarity-dependent resistance that can also depend on the sign and nature of the charge carrier. The bulk charge mobility along z in the NF may also depend on field direction. Beyond this there will be a variety of charging effects due to the linear coupling of P and flow. If we consider, for example, steady, incompressible nematic laminar flow, then the director is generally nearly parallel to the velocity Complex flows will thus produce complex patterns of polarization charge. Reorientation of P is dis- , and, if driven by electric field, gives a highly anisotropic contribution to the net electric conductivity, $!"+"$ ! *!1",-)"""! P , and $""#+".",-)"""|| P [54]. For RM734 $! ~ 10 -3 /.cm, which is in the semiconducting range. Under these circumstances accumulation of one sign of charge in the fluid can occur when an applied AC field gets out of phase with polarization reversal. Additional inherent asymmetries, such as differences in mobility or chemical character between positive and negative ionic impurities, or an intrinsic tendency for splay distortion of the P(r) field itself, can also contribute. RM734 molecules (SI Appendix Fig. S18) with periodic boundary conditions, equilibrated in the NPT ensemble at p = 1 atm for a range of temperatures spanning the N and NF phases, using the APPLE&P force field [58], which has been successfully applied in previous studies of 5CB nematic [59], and CB7CB twistbend [60] phases. More details of the simulations can be found in SI Appendix Sec. S6.

Atomistic Molecular Dynamics (MD) Simulation
The simulations probe the equilibration of RM734 in two distinct condensed LC states: pending on the orientation of its u (along +z or -z, respectively). These states represent the equilibrium limits of polar order and polar disorder in the simulation volume.

Figures 7B-E
show the resulting positional pair-correlation functions, gP(",z) and gNP(",z) of the equilibrated POL and NONPOL systems, where for the NONPOL system gNP(",z) = gNP par (",z) + gNP anti (",z) is the sum of the correlations between the molecular pairs with relative parallel or antiparallel orientations of u. The g(",z) are ✦ ✲ averaged conditional probability densities of molecular centers around a molecule with its center at the origin and long axis along z, and thus are uniaxially symmetric in (",✦,z✮ cylindrical coordinates, reflecting the uniaxial symmetry of the N and NF phases. They all exhibit a molecule-shaped, low-density region (g(",z) ~ 0) around the origin from the steric overlap exclusion of the molecules; an asymptotic constant value at large " giving the normalized average density (g(",z) = 1); and distinct peaks indicating preferred modes of molecular packing. If we were to consider a similarly equilibrated system of rods that are marked "O" and "H" on their ends but are otherwise symmetric (e.g., hard spherocylinders marked as either OH or HO) then we would find identical g(",z): gP(",z) = gNP(",z) = 2gNP par (",z) = 2gNP anti (",z). This motivates the comparison in Fig. 7, in which the g(",z) of RM734 display a number of striking features that directly exhibit the effects of its structural and electrostatic polarity on the packing of neighboring molecules.
The POL gP(",z) in Fig. 7B shows the equilibrated, local molecular packing preference in the limit of polar order, i.e., in the system with the maximum number of contacts between like-oriented molecules.
Its prominent features are on-axis sharp arcs at (" = 0, z = ±22 Å), indicating on-average coaxial molecular association into polar chain-like (OH-OH-OH) associations having a center-to-center spacing along z of the molecular length, 22 Å, stabilized by the electrostatic attraction of the nitro and methoxy ends of the molecules (Fig. 7F); and off-axis peaks at " = 5 Å, z = ±6 Å, indicating polar side-by side association (Fig.   7G). These correlations indicate that specific electrostatic interactions between oppositely charged groups on the two molecules (e.g., between positively charged terminal or lateral methoxy H atoms and negatively charged nitro O atoms) play a dominant role in stabilizing such pair configurations.
The NONPOL system enforces the maximum number of molecular contacts between molecules of opposite orientation. In this situation of maximum polar disorder, possible equilibrated molecular correlations could range from being (i) dominantly antiparallel end-to-end (e.g., OH-HO-OH chains, with sideto-side polar correlations, as in the bilayer smectics of strongly polar molecules [61]); to being (ii) polar end-to-end (a mixture of OH-OH-OH and HO-HO-HO chains with the OH-HO interactions side-by-side).
RM734 is distinctly in the latter category, as, remarkably, the principal polar ordering motifs of

Materials and Methods
Synthesis of RM734 -4-[(4-nitrophenoxy)carbonyl]phenyl 2,4-dimethoxybenzoate (RM734, Fig. 1A) is a rod-shaped mesogen first synthesized by Mandle et al. [30]. It was reported to have an isotropic (I) phase and two additional phases with nematic character, with transition temperatures as follows: I -187 ºC -N -133ºC -NX -X. Our preparation is based on general synthetic reactions and uses procedures only slightly modified from those described in the literature cited. A nanoscale volume containing 384 molecules is equilibrated in these simulations into two distinct LC states: a POLAR system with all polar molecular long axes, u, along +z, and a NONPOLAR system with half along +z and half along -z. Equilibration of the molecular conformation and packing is readily achieved but end-to-end flips are rare so the equilibrated states remain in the limit of polar or non-polar nematic order, respectively. (B,F,G) The POL simulation shows directly the dominant pair correlations adopted by molecules that are polar ordered, in the form of conditional probability densities, g(",z), of molecular centers around a molecule with its center at the origin and long axis u along z. The g(",z) are ✦ ✲ averaged to be uniaxially symmetric, reflecting the uniaxial symmetry of the N and NF phases. They exhibit a molecule-shaped, low-density region (g(",z) ~ 0) around the origin resulting from the steric overlap exclusion of the molecules; an asymptotic constant value at large " giving the normalized average density (g(",z) = 1); and distinct peaks indicating preferred modes of molecular packing. This analysis reveals two principal preferred packing modes in the POL system: (B,F) polar head-to-tail association stabilized by the attraction of the terminal nitro and methoxy groups, and (B,G) polar side-by-side association governed by group charges along the molecule, nitro-lateral methoxy attraction, and steric interactions of the lateral methoxys. (D,E) The NONPOL system exhibits distinct correlation functions for antiparallel and parallel molecular pairs, gNP anti (",z) and gNP par (",z). (E,H,I) The preferred antiparallel packing gives strong side-by-side correlations, governed by group charges along the molecule; and (E,J,K) weaker antipolar nitro-nitro end-to-end association. (D,F,G) The parallel correlations in the NONPOL system are the most relevant to the stability of polar order in the NF phase, as they are determined by the inherent tendency of the molecular interactions for polar ordering in the presence of enforced polar disorder. Comparison of (B) and (D) shows identical preferred modes of parallel association in the two systems, with the POL system correlations being even stronger in the NONPOL system. This  Section S2 -PLANAR-ALIGNED LARGER-AREA CELL TEXTURE (Fig. S4) (Figs. S5-S10) . 7 Section S4 -COMBINING PPR AND PND DOMAIN WALLS (Fig. S11) . 16 Section S5 -SPLAY IN THE N AND N F TEXTURES (Fig. S12) . 17 Section S6 -SANDWICH CELL: EO BEHAVIOR & THE FRÉEDERICKSZ TRANSITION (Figs. S13,S14) .. 18

Observations of Response to Applied Electric Field -
Section S7 -CHARACTERISTICS OF FIELD-INDUCED FLOW (Figs. S15,S16) (Fig. S17) . 23 Section S9 -ATOMISTIC SIMULATIONS -METHODOLOGY (Fig. S18) (Fig. S1) is a rod-shaped molecule about 20 Å long and 5 Å in diameter, with a longitudinal electric dipole moment of about 11 Debye. First reported by Mandle et al. [1,2], it was found to melt at T = 140ºC and have an isotropic (I) phase and two additional phases with nematic-like character, with transition temperatures on cooling as follows: I -188ºC -N -133ºC -NX. Our synthetic scheme, shown in Fig. S1, is based on general synthetic reactions and procedures only slightly modified from those described in the literature cited. The synthesized compound has very similar phase transition temperatures to those reported by Mandle (Fig. S2).

4-(2,4-dimethoxybenzoyloxy)benzoic acid (4)
-A solution of the benzyl-protected carboxylic acid (3) (9.4g, 24 mmol) in CH2Cl2 (100 mL) and EtOH (100 mL) was first evacuated and purged with argon, then 10% Pd/C catalyst (2 g) was added. The argon atmosphere was replaced by hydrogen gas, and the reaction mixture was stirred at room temperature for two hours. Hydrogen was pumped out of the system and the flask was purged thoroughly with argon. The mixture was filtered through Celite and the solvents were removed under reduced pressure. This gave product as a white powder (6.99 g 96%).

4-[(4-nitrophenoxy)carbonyl]phenyl 2,4-dimethoxybenzoate -
To a suspension of compound 4 (4.38 g, 15 mmol) and 4-nitro phenol (5) (2.02 g, 15 mmol) in CH2Cl2 (125 mL) was added DCC (6.00 g, 30 mmol) and trace of DMAP. The reaction mixture was stirred at room temperature for three days, then filtered, washed with water, 5% CH3COOH, water, and brine, then dried over MgSO4, filtered, and concentrated at reduced pressure. The resulting product was purified by flash chromatography (silica gel, CH2Cl2/2% EtOAc) followed by crystallization from CH3CN. The product is a white crystalline solid (3.92 g 62% Differential scanning calorimetry measurements of RM734 are shown in Fig. S2. Figure S2: DSC measurements of RM734. The heating and cooling rates were respectively ±10°C/min. The heats of transition were computed using a molecular weight of 423.38.  Figure). As the cell thickness was much smaller than the gap between the electrodes, a voltage applied between the two electrodes produces a complete uniform alignment of the ferroelectric polarization P in the gap center. The cell is seen in perspective above. (B) Capillary geometry. Two copper wires 0.55 mm in diameter with flat ends were inserted into coaxial position inside a cylindrical glass capillary of slightly larger diameter filled with liquid crystal. The gap between the ends of the electrodes was 0.2 mm.

Methods -Electrooptic Measurement -
The liquid crystal phase sequence and phase transition temperatures were determined by polarized light microscopy using a Nikon Eclipse E400 POL microscope equipped with an Instec STC200 temperature-controlled hot stage. DSC was performed using a Mettler DSC823e differential scanning calorimeter. Electrooptic measurements were made using a Zeiss microscope, a Spectra Physics 105-1 HeNe laser, an EZ Digital FG-8002 function generator, and a Tektronix TDS 2014B oscilloscope.
Unless otherwise noted, optical textures and domain structures of RM734 reported in the paper were observed in a t = 11 µm-thick LC Vision cells having an inplane field geometry comprising a pair of 5mm long ITO strip electrodes spaced by d = 1.04 mm. The plates had buffed polymer alignment in the direction of the field (normal to the electrode gap), which gave a weak anchoring alignment of the director normal to the gap and along the field.
Polarization Measurement -Polarization was measured by time integration of the current through two-terminal LC cells in which saturated polarization reversal was achieved. In-plane field cells, such as that in Fig. S3A, and the" +$6*%7/*+869" +.8:*86" (%7;&.;(%7" (6(+&/.7(" <68))" +8=*6; 68/$"+(66"in is generally aligned along the rubbing direction z, producing good extinction, . Experiments with small probe fields applied along the rubbing direction confirm that the average P(r) is directed toward the right. Vertical white lines show the electrode edges. (B) Rotating the cell reveals the pink birefringence color of the aligned domains (in the third-order Michel-Levy band). An applied field, E = +2 V/cm parallel to P, stabilizes the zero-field orientation. (C,D) Reversing the field to E ~ -0.5 V/cm begins the reorientation process, with bands in the texture corresponding to alternating rotation directions of the n,P couple. (E,F) At E ~ -1 V/cm the n,P couple has rotated by !! ~ ±90º, forming regions of different signs of rotation separated by boundary walls. These regions are birefringent rather than extinguishing because n(r) twists non-uniformly across the thickness of the cell. (G,H) At E ~ -2 V/cm, the rotations of the n,P couple are nearly +" or -", with the regions of opposite rotation now separated by narrow 2" walls. This is immediately followed by a process in which domain boundaries nucleate holes whose growth leads to the elimination of the domain walls and the formation of large regions of uniform n and P, a state that is identical to that of (A) but with P reversed. This polarization reversal scenario takes place for fields|E| < 2 V/cm and is repeatable. Scale bar = 200 µm. " Figs. 4, S5-S11, the optical texture of the N phase in our planaraligned cells is locally smooth away from the transition. This texture roughens upon cooling toward the NF phase, developing a random pattern of micron-size domains extended along n, which, once in the NF phase, coarsen into a smooth texture very similar to that of the nematic, but marked by a pattern of smooth lines, some of which extend for long distances generally parallel to n, and others which form closed loops to make macroscopic, lens-shaped domains, as seen in Figs. 1,S6,S7,S11. The experiments presented in Figs. 1,S5,S11 show that these lines demarcate areas of opposite polarization density, all formed in the absence of applied electric field. This remarkable evolution and the response of these domains to applied field are at the heart of the phenomenon of ferroelectricity, giving rise to its characteristic behaviors of macroscopic polar ordering and hysteretic dielectric behavior. Generally, the domain walls are visible because they have a width comparable to the cell thickness, and they appear to have an intrawall director with a boundary condition that tends to keep n locally parallel to the wall (Fig. 6A). A resulting feature of these walls, evident in Figs. 1,S5,S11, is that wherever they run exactly parallel to n(r), they lose optical contrast and can no longer be seen in DTLM. This means that at these places there is no reorientation of n(r) upon crossing the domain boundary: n is in the same direction inside and outside of the lens-shaped domains and through the wall, whereas P changes sign. We refer to these as pure polarization reversal (PPR) lines. PPR lines show that the differing states of the low-temperature phase found across domain walls differ only in the sign of P, and that P is therefore a suitable order parameter for describing the transition, i.e., that the NF phase is a proper ferroelectric. PPR surfaces form upon cooling through the N-NF transition as the correlated volumes having predominately a particular sign of P(r) grow in size and the polarization fluctuations become slow and glassy. The resulting three-dimensional domains ultimately establish an equilibrium degree of order, encoded in the magnitude of P, and confine the transitions to the opposite sign of P in neighboring regions into PPR surfaces of a local structure that includes an internal nematic sheet, the central-surface of the wall where P actually changes sign. In thin cells, these PPR walls have the appearance of lines but they are of course three-dimensional, with a structure controlled in part by the aligning surfaces of the cell. In a bulk 3D system, the PPR walls must form some sort of closed-shell structures.

Ferroelectric domains -As evident in
With this observation, we now have two distinct kinds of defect lines in our cells where P(r) can change sign relative to the lab frame, z: the PPR lines in which P(r) flips by 180º but n(r) is unchanged, and the "disclination (PnD) lines of Figs. 2 and S6, in which P(r) and n(r) rotate together, maintaining a fixed relative orientation. Observations of the PPR lines show that they avoid making angles " relative to z larger that about 45º. This is seen, for example, in the lens domains where the PPR line is forced to be a closed loop. The line, rather than forming a continuously circular domain wall, where " would at two locations get to be 90º, instead jumps from " ~ -45º to " ~ +45º in order to complete the loop. Similar behavior is also shown in SI Appendix Fig. S6 for an extended line, where such a jump is maintained until " can be become sufficiently small without it. "-PPR and "-PnD lines can combine to make 2" walls in the uniquely ferroelectric way shown in Fig. S7.

Figure S5:
Step by step through the life of a lens-shaped NF domain in an applied reversal field increasing in the range 0 < E < 1 V/mm. (A) Yellow arrows show the director n, green the applied field E, and black the polarization, P. Virgin domains before the field is applied. The director at the domain wall tends to follow the wall, otherwise n is uniform within each domain. The domain wall is invisible at the widest part of the lens, indicating that the wall does not involve reorientation of n, only reversal of the polarization -a pure polarization reversal (PPR) wall. The lens shape of the domain is due the interaction of n inside the wall with the surface buffing, making PPR lines stable only for " <~ 45º. (B) In a small applied field (E ~ 0.1 V/mm), the domain interior reorients through an azimuthal angle !#~ 90º. The area outside the domain, does not reorient, indicating that it is already aligned with the field. (C) Reorientation approaches 180º. (D-F) A reorientational disclination loop nucleates in the cell mid-plane to accommodate the induced 180º reorientation of n, and moves in response to the applied field to surround the second domain. (F,G) The domain shrinks as the disclination loop collapses. The first lens domain also responds to the field but maintains its shape, its boundary apparently trapped more strongly. (H) Cross-sectional sketches of the domain structure before and during field application. The initial domain boundary is a PPR polarization wall with no disclinations in n. In the region between NF(+) and NF(-) , the system must pass through P = 0, i.e., be nematic. The field-induced reorientation inside the disclination loop (green) adds an additional 180º rotation (a polarization-director disclination, PnD) to the domain wall, as well as surface twist disclinations inside the loop which connect the bulk fieldinduced reorientation to the surfaces. The green swoop indicates the field-induced reorientation of the domain interior, causing the formation of a compound "-PPR / "-PnD boundary to make a 2" wall. The reorientation on the surface is mediated by " twist disclinations (magenta). Scale bar = 100 µm.

Modeling the N-NF phase transition -
The low-temperature phase of RM734 has been proposed to be a "splay nematic" phase [3,4,5], a splay-bend ground state of the sort initially described theoretically by Dozov [6 ], in which the packing frustration of electrically and structurally polar (pear-shaped) molecules is relieved by the spontaneous adoption of local splay deformation, with 3D space filled by planar (x,z) sheets of splay (✦ny/✦y) alternating in sign of splay and therefore also in sign of polarization (see Ref. [4], Fig. 11). However, we find that: (i) the NF develops a large, macroscopic polarization as it grows in fieldfree through the transition from the N phase, and (ii) the ultimate polarized state at low temperature is effectively saturated, exhibiting no other significant birefringence or internal structural change upon cooling in the NF phase and with little response to large applied fields tending to increase P (i.e., E parallel to P). These conditions appear to preclude any significant cancellation of P by opposed splay domains. We propose that the ground state of the NF phase has, aside from fluctuations, uniform P and that it is a proper ferroelectric phase, where P is the principal order parameter and phase stabilityoriginates from a combination of short-range interactions that favor neighboring molecular long axes to be parallel and in the same polar orientation (Fig. 1), and long-range dipole-dipole interactions. Under this condition, the N-NF transition would be viewed as Ising-like, with the N phase being a collection of molecules with long axes on average parallel to n but with disordered polar directions (<pi> = 0), and the NF phase also having long axes on average parallel to n but with polar order N<pi> = P. Ignoring for the moment the dipole-dipole interactions, with only ferroelectric short-range interactions this transition would be predicted to be second-order with 3D critical exponents of the Ising universality class [7].
Mertelj et al. have observed pretransitional effects, including the softening of the splay elastic constant, K s , and a diverging dielectric anisotropy ✧ in the N phase, that indicate growing polarization correlations, and flexoelectric coupling between these fluctuations and director splay as the transition is approached from above, observations which they have interpreted using a Landau-de Gennes model [4,5,8]. . This free energy is consistent with the NF phase having uniform P, with a weak tendency for splay that is suppressed by the polarization charge effects discussed above, and by the energy cost of the defects required to introduce splay modulation.
An additional pretransitional effect not yet adequately described is the striking anisotropy of the pretransitional and coarsening correlations in the fluctuations of P through the transition (Fig. S8, S9), with the domains becoming increasingly extended along z, until they end up as polar regions separated by PPR walls running largely parallel to z. Given the significant anisotropies introduced into the NF phase by space charge effects, it is natural to consider that the polarization charge energy UP associated with electric dipole-dipole interactions may also affect the phase transition and pretransition fluctuations. The resulting interaction energy combines short-ranged ferroelectric and long-ranged dipole-dipole forces.
The critical behavior of such systems has been studied extensively in an effort to understand certain magnetic materials that have short-range ferromagnetic exchange forces, but where the long-range dipolar interactions are also important [9,10,11]. In these systems, short-ranged interactions are included in a model Hamiltonian as nearest-neighbor Ising or Heisenberg-like, and the long-ranged interactions are calculated explicitly. Renormalization group analysis shows that the long-range interactions make the magnetic correlations dipolar-anisotropic near the transition in the high temperature phase [12,13], extending them along z by strongly suppressing longitudinal charge density (✦Pz/✦z) fluctuations [9,10]. Specifically, starting with the free energy expression Eq. (1) from ref. [5] and adding the dipole-dipole in-teraction term UP from above, the structure factor for fluctuations in Pz becomes !Pz(q)Pz(q)*" The dipole-dipole (third) term produces extended correlations that grow as ✩ ($) along x and y but as ✩ ($) 2 along z [10], suppressing ✫ (q) for finite qz as is observed qualitatively from the image sequences of the textures upon passing through the phase transition, and from their optical Fourier transforms in Fig. S9. Because of this anisotropy, the correlation volume in this model grows in 3D as V ~ ✩ ($) 4 rather that the isotropic V ~ ✩ ($) 3 , reducing the upper marginal dimensionality of the transition to 3D, making the transition mean-field-like with logarithmic corrections, rather than fluctuation-dominated with 3D Ising universality [14]. 15 Figure S6: Typical DTLM images of the t = 11 µm, planar-aligned, in-plane field cell with E = 0. (A) N phase (T = 140ºC). (B) NF phase (T = 120ºC). The director n is generally along the buffing direction z, here making an angle of about 45º to the crossed polarizer and analyzer but giving good extinction when rotated so that n is along the polarizer or analyzer direction. The linear defects in the texture are " director disclinations and surface memory reorientations [15] Figure S5B during the coarsening (Figures 1,S7,S8). Ferroelectric polarization directions are shown in (A). Within the interior of the wall, n is always parallel to the wall direction, whereas outside of the wall n has its macroscopic orientation along the buffing (yellow line), maintained by polarization charge stiffening. (A!C) This difference makes increasing the angle " between the wall and n increasingly costly in energy, creating a line tension that shortens wall segments as " increases. T = 120ºC. Scale bar = 100 µm. ! N-NF phase coexistence upon melting the NF -Slow heating of RM734 through the NF-N transition produces a distinctive melting process in which the nematic appears as extended, flexible filaments in a background of NF. Examples of these filaments are shown in Fig. S10, where a nematic front is moving in from the right as the temperature increases slowly. The filaments, which have diameters are in the range 2 µm < Df < 10 µm, grow in from the right, where their density is largest. The nematic filaments penetrate the NF phase but do not otherwise disturb it, with the NF maintaining its alignment and birefringence independent of any nearby filaments. The nematic alignment inside each filament appears to be uniform, with the director oriented parallel to the filament axis, so that the filaments can generally be seen with high contrast, except where they are aligned parallel to the NF background field. As with the domain walls in the NF phase, the nematic filaments lose optical contrast when oriented parallel to the director field of the background NF phase. The orientation of the filaments does not appear to be strongly coupled to the NF director field and varies greatly across the sample. These filaments show that the N and NF phases can coexist as distinct states, at least metastably&## Figure S10: Filamentous domains of the N phase moving into the NF phase upon heating through the N-NF transition near T = 130 ºC in a t = 11 µm cell. The N filaments grow in from the right. The filaments lose their optical contrast when they are oriented parallel to the director of the background (orange) NF phase, indicating that, as in the PPR walls, the director inside the filaments is oriented parallel to the filament boundary. This phase coexistence is evidence that the N-NF transition is first order. Scale bar = 100 µm.

SECTION 4 -COMBINING PPR AND PND DOMAIN WALLS
The combination of "-PPR and "-PnD lines creates a novel hybrid defect line unique to the NF phase. Fig. S11A shows two PPR lines separating domains of opposite P, with a small field beginning to reverse the central domain. The PPR lines are barely visible, as noted in Fig. S7, but increased field strength reverses the central domain, making it visible and leaving unswitched sheets ~'P thick at the surfaces. Then, at sufficiently high field, in a process analogous to the "order reconstruction" of confined splay in a nematic [16,17], pink holes open up in these sheets (Fig. S11B). This nucleation and the resulting movement of "-PnD lines to the boundaries of the central domain, leave it completely pink, i.e., uniformly switched with P now along E, and now identical in structure to the outer area, separated from it by the hybrid "-PPR/ "-PnD 2"-walls, now visible because of their PnD component. The cross-sectional structure in Fig. S11C shows the 360º reorientation of P on passing through the wall and that in an applied field, such walls have energy in excess of that of the surrounding areas. This energy difference increases with field, which confines the 180º PnD reorientation of the wall to a central sheet of width 'E = $K/PE, a high energy director configuration in a sea of uniform orientation.
At sufficiently high field, "order reconstruction" holes [16,17] will nucleate in the sheet, inducing gaps in the wall such that it will recede (Fig. S11D-G). This process is one-way: once the wall recedes, the area vacated shows no memory of it when the field is reversed. As a result, the PPR lines in a newly cooled cell tend to disappear by nucleation and movement of gaps when exposed to a changing field.
The comparisons above of polar vs. dielectric response to field also apply to this process. in, and shrinking of, 2"-PPR/PnD walls, induced by increased field. Since the walls have higher energy than the uniform state that replaces them, they are squeezed out by the field (blue arrows) and do not regrow upon decreasing the field, a hysteretic response. (H) A second field reversal produces a bend instability in the director field like that shown in Figure 5. The sections in C and D show the 2"-PPR/PnD wall structure and its squeezed out remnant respectively. Scale bar = 100 µm.

SECTION 5 -SPLAY IN THE N AND N F TEXTURES
The evolution of the cell upon cooling into the NF phase, ending, apart from the presence of the domain walls in Fig. 1, in textures that are optically smooth with close to the same birefringence as the nematic, suggests that there are no sub-resolution orientational modulations or non-uniformities that would reduce the birefringence of the equilibrium n(r),P(r) fields of the NF phase beyond those of the N phase. However, the polar symmetry of the NF phase implies a polar orientational distribution of the molecules, which, because of the lack of reflection symmetry of their steric shape, should generate a tendency for director splay in the NF phase that is correlated with the direction of P(r), like that of flower stems in a bouquet, with P(r) indicating the growth direction [18]. This effect has been observed in calamitic chiral smectics [19], and in polar-ordered smectic bent-core B2 phases that are ferroelectric (SmCSPF) [20]. The latter break up into structures of linear splay domains, an instability which generates the B7 phases, for example [21]. Such structures appear only above a threshold value of preferred splay amplitude because filling 3D space with splay always requires some sort of non-splayed or oppositely-splayed defects that cost extra energy [22,23]. Below this threshold, the splay is simply frustrated in domains of uniform orientation. In some situations, however, splay is required, such as in the n(r),P(r) field around the air bubbles in Fig. 5C and Fig. S12, and around and inside the P-reversal domains in Fig. 1D and Fig. S5. Here different signs of polarization splay are required to make the two 180º wedge defects on opposite sides of the bubble (red dots), and at opposite ends of the polarization domains. By symmetry, these opposing structures must be different. However, they are in fact very similar in appearance, implying that the preference for a particular sign of splay does not significantly affect their structure. The NF phase therefore appears to have large polarization but a weak tendency for splay, indicating a small splay flexoelectric coefficient [24] that is perhaps a result of its molecular-scale organization, discussed in the text and in Section S10. Figure S12: DTLM images of textures around an air bubble in the t = 11 µm cell. Orientational anchoring at the bubble /LC interface forces the director n to be tangent to the interface, so that it bends going around the bubble perimeter. (A) A 30 µm diameter bubble in the N and NF phase. (B) 35 µm diameter (left) and 60 µm wide (right) bubbles in the NF phases. Comparing the N and NF textures shows that the tangential surface anchoring is stronger in the NF phase and that the director bend persists to much larger distances from the bubble surface. This is due to the competition between strong polarization charge-generated splay rigidity and normal bend rigidity in the NF phase, making splay deformations very gradual. Here " wedge disclinations (red dots in the sketch) stabilize required splay near the bubble. !

SECTION 6 -SANDWICH CELL: ELECTRO-OPTICS & THE SPLAY-BEND FRÉEDERICKSZ TRANSITION
The NF phase exhibits characteristic high-polarization ferroelectric LC behavior, illustrated in Fig. S13, which shows a sequence of DTLM images of RM734 in a conventional, transparent capacitor cell t = 4.5 µm thick, with antiparallel-buffed polyimide-coated ITO electrodes that give unidirectional planar alignment with a few degrees of pretilt. The sample temperature is held near the N-NF transition, a small, inplane temperature gradient ensuring that part of the sample is in the NF phase while the rest is in the N phase, with a wide transition region (demarcated by white and black dashed lines in Fig. S13A) where the two phases form overlapping wedges (sketched in Fig. 8C) and the birefringence is lower. In the N phase, the LC responds dielectrically to an applied 1 kHz triangle wave voltage of amplitude Vp, resulting in a decrease of birefringence (n, beginning at Vp ~ 1.5 V. For Vp ~ 3 V, the birefringent optical retardance (nt is reduced from 900 nm to 300 nm [25], corresponding to )(Vp) increasing from#) ~ 0º to ) ~ 54º, a substantial rotation of n toward the field direction. However, the net birefringence increases continuously through the wedge region as the thickness of the N layer decreases (Fig. S13C). In contrast, the part of the cell in the NF phase exhibits no observable change in birefringence in this voltage range. This behavior can be understood on the basis of the "block polarization" model developed to explain V-shaped switching in high-polarization FLCs [26,27,28]. In this model, polarization charge induced by reorientation of P completely cancels E in the LC and the polarization direction is electrostatically controlled. Insulating layers without reorienting polarization at the LC/glass interfaces such as the polymer alignment layer, charge depletion in the ITO, and bound polarization at the surface, are accounted for as capacitive elements with a net capacitance per unit area, C. The orientation of the polarization field (V) with these assumptions is given by sin)(V) = V/Vsat, where Vsat = P/C [26,27,28], showing that when P is large, a large applied voltage is required to achieve substantial reorientation of n,P. This NF cell can be compared with the bent-core SmAPF ITO electro-optic cell of Ref [28], where a saturation voltage Vsat = 15 V was measured for material with P = 0.85 µC/cm 2 , leading us to expect Vsat ~ 100 V for LC in the NF phase with P ~ 6 µC/cm 2 in a cell with 50 nm-thick insulating layers with * = 5 at the two cell boundaries. It is therefore not surprising that the NF phase shows very little response in the applied voltage range explored here. In a further experiment, we cooled through the N-NF transition while continuously applying a Vp ~ 16 V triangle wave, observing Fréedericksz-type reorientation to a homeotropic state in the N phase and the reimposition of planar alignment by polarization stabilization in the NF phase (see Fig. S14). Figure   The N phase with the field applied starts at ) ~ 85º (above threshold and nearly homeotropic) at T = 140ºC, but, approaching the transition, (C) ) decreases to ) ~ 80º near T = 131ºC, where (D) the NF phase comes in via irregular domain boundaries that (E) anneal into an ordered ) ~ 0º planar-aligned NF domain (orange birefringence color). However, (F-G) as T is lowered to T = 120ºC this field induces a defected pattern of increased ), a birefringent texture that is permanently written when the field is removed. This process must involve breakdown of the interfacial capacitors discussed in the text. Texture (H) is obtained with a Vp ~ 16 V, 1 Hz triangle wave. Scale bar = 1 mm.!

! SECTION 7 -TEMPERATURE DEPENDENCE OF FIELD-INDUCED FLOW
" highest is in the NF and is flowing. (C) T = 140ºC. The LC is now entirely in the N phase, in a Schlieren texture, and there is no observable flow, showing that the defect motion and flow is a feature of the NF phase (Fig. 6). Scale bar = 1mm. See also  Figure S16: Reflected light microscopy image of the cell in Fig. S15, a t = 10 µm thick planar aligned cell of RM734 between untreated glass plates with two evaporated gold electrodes on one of the plates, separated by a d = 60 µm gap. This image is taken at T = 120 ºC at the instant of field reversal, where the resulting polarization reversal generates a periodic array of bend domain walls locally normal to the reversing E and to the director field n(r), as in Fig. 5A. At the top of the image the lines approach being radial as in Fig. S15A. This image shows a remarkable feature of the NF electrooptics, specifically that there is weak applied field that penetrates over the electrodes that is strong enough to produce polarization reversal in the LC. The yellow lines indicate typical director field orientations.  Figure S18: (A) Instantaneous configuration from the POL MD simulation system of RM734 at T = 130 °C, employing a polarizable molecular model. The system was initialized in a polar state, and equilibrated retains a high degree of nematic and polar orientational order, as end-to-end flips are not observed. The vertical cell dimension is 70Å. # tensor >6?, and the biaxial order parameter B is defined as the difference between the two smallest eigenvalues. Polar order is assessed by measuring the (vector) polar order parameter @ 7 ! " 8 : % " %&! , from which a scalar polar order parameter A 7 >@? can be obtained.
For the POL simulation of the polarizable model at 130 °C, we measure a large nematic order parameter, S = 0.787 ± 0.009, and a nearly saturated polar order parameter, # = 0.924 ± 0.003, with negligible biaxiality (B = 0.013 ± 0.002). Moreover, the polar order parameter P is co-linear with n, the principal eigenvector of >6?. Given that there appear to be no long-range positional correlations (as shown below), the simulated state appears to be a uniaxial ferroelectric nematic (NF) phase.
It is interesting to compare these results with those from the NON-POLAR simulation under the same conditions (T = 130 °C, polarizable molecular model). The nematic order parameter in this case is S = 0.782 ± 0.018, quite similar to that of the polar system, while the polar and biaxial order parameters are small (P = 0.013 ± 0.004, B = 0.028 ± 0.003), as expected for a conventional uniaxial nematic (N) state. The fact that the magnitude of S is nearly the same in the POL and NONPOL states is generally consistent with the experimental observation that the birefringence does not change significantly through the N -NF transition. Simulated mass density at T = 130ºC is ! = 1.33 g/cm 3 .
Ferroelectric polarization density -The measured spontaneous ferroelectric polarization density P in the NF phase of RM734 is large, increasing with decreasing temperature below the N -NF transition to a saturation value of around P = 6 µC/cm 2 (Fig. 3). This implies a high degree of polar order in the NF phase, that can be further quantified by comparison with the polarization A ' computed by simulation.
As noted above, RM734 has a large electric dipole moment p = 11.4 D, as determined from quantum chemistry calculations at the B3LYP/6-31G* level of theory (Fig. S19). Higher-level quantum chemistry calculations were used to assign site charges to atom sites and lone pair electron sites in the molecular mechanics model used in our atomistic simulations. The resulting static site charges, shown in Fig. S20, are also consistent with a molecular dipole moment of around 11 D (computed using polarization # , where ri and qi are the site positions and charges, and the sum ranges over all n sites in the molecule). Note that the dipole moment has a weak dependence on molecular conformation, and that molecules sample an ensemble of low-energy conformations over the course of a simulation. For the POL simulation of the polarizable model at 130 °C, we measure an average static molecular dipole moment (from static site charges) of magnitude >I *+,+(-? = 11.24 ± 0.01 D. For the polarizable models, there is also an induced molecular dipole moment component, which has an average magnitude of # >I ()./-0. ? = 1.46 ± 0.02 D, but has a nearly isotropic orientational distribution, so the average magnitude of the total molecular dipole moment (the sum of static and induced contributions) is nearly equal to the static contribution, >I +1+,2 ? = 11.20 ± 0.01 D. The fact that the induced molecular dipole has a nearly isotropic orientational distribution is a consequence of the boundary conditions, which ensure that the average electric field is zero at any point in th" e system (there is no bound charge at the surface of the system, so the depolarization field vanishes), so the average magnitude of the induced dipole moment vector is close to zero, J>B ()./-0. ?J = 0.053 ± 0.009 D.
We can gain further insight by resolving the total ferrolectric polarization density into contributions from specific dipolar groups. To accomplish this, we employ a unique decomposition of charges into elementary charge-neutral dipolar groups (bonds and rings), as shown in Fig. S21. Group dipoles can be further aggregated into functional groups, which are indicated by color coding in Fig. S21, together with the dipole moments and average contributions to ferroelectric polarization density PS of specific functional groups. The terminal nitro group and the ring to which it is attached are highly dipolar, and make a dom-! Figure S21: Decomposition of the static site charge distribution into group dipole contributions. Irreducible bond and ring dipole moments are shown as small arrows, where the numerical value is the dipole moment in Debye (D). The dipole moments of specific functional groups are also indicated (large arrows and large, non-italic text). The numbers in italics are the average contributions of specific functional groups to the computed ferroelectric polarization density PS, in units of #C/cm 2 . The nitro group and the ring to which it is attached (the nitro ring) have the largest dipole moments, and together contribute 64% of the total polarization density. Four functional groups (nitro, nitro ring, central ring, and terminal methoxy) contribute 90% of the total polarization density. The ester groups and lateral methoxy possess substantial lateral dipole moments, which may contribute to intermolecular association. ! inant (~ 64%) contribution to PS. Four functional groups (nitro, nitro ring, central ring, and terminal methoxy) account for ~ 90% of the molecular polarization density.
We have calculated the average ferroelectric polarization density of the maximally polar equilibrium state of the POL simulation at 130°C from >@ ' ? 7 > ?, where V is the system volume. For the POLAR simulation of the polarizable model, the polarization density magnitude A ' 7 J>@ 4 ?J = 6.17 ± 0.01 µC/cm 2 , where only 0.13 ± 0.03 µC/cm 2 is due to the induced polarization. This calculated PS is in quantitative agreement with the saturation polarization density P measured experimentally (Fig. 3). This value is also nearly the same as that obtained from the simple estimate given in the text assuming 100% polar order, as might be expected given the high value of the POL system polar order parameter given above, # = 0.924. The remanant orientational disorder is due to small orientation fluctuations about n, basically all that is allowed in the POL system. Consequently, # depends only weakly on T, with the MD giving A ' = 6.170 ± 0.008 µC/cm 2 at T = 130ºC and A ' = 6.368 ± 0.002 µC/cm 2 at T = 110ºC.
An important inference of this agreement with the NF phase experimental value is that at low T RM734 essentially becomes the POL system, i.e. is a polar nematic with no molecular flips, and remnant polar disorder that is strictly short ranged (~few molecule) small angle orientation fluctuations about the director. At higher temperatures P decreases in the NF phase because of the growth of longer length scale fluctuations and disordering modes. But these disappear upon cooling to the saturated state at low T, where the fluctuations in the NF become consistent with those allowed in the nanoscale volume and periodic boundary conditions of the POL simulation. On the other hand, the simulation volume is chosen to be large enough to observe the local molecular packing driven by specific intermolecular interactions, discussed next. Figure S22: Results of atomistic molecular dynamic simulations designed specifically to explore the molecular interactions and resulting positional /orientational correlations responsible for the polar molecular ordering of RM734, shown in (A). A nanoscale volume containing 384 molecules is equilibrated in these simulations into two distinct LC states: a POLAR system with all polar molecular long axes, u, along +z, and a NONPOLAR system with half along +z and half along -z. Equilibration of the molecular conformation and packing is readily achieved but end-to-end flips are rare so the equilibrated states remain in the limit of polar or non-polar nematic order, respectively. (B,F,G) The POL simulation shows directly the dominant pair correlations adopted by molecules that are polar ordered, in the form of conditional probability densities, g(",z), of molecular centers around a molecule with its center at the origin and long axis u along z. The g(",z) are ✦ ✲ averaged to be uniaxially symmetric, reflecting the uniaxial symmetry of the N and NF phases. They exhibit a molecule-shaped, low-density region (g(",z) ~ 0) around the origin resulting from the steric overlap exclusion of the molecules; an asymptotic constant value at large " giving the normalized average density (g(",z) = 1); and distinct peaks indicating preferred modes of molecular packing. This analysis reveals two principal preferred packing modes in the POL system: (B,F) polar head-to-tail association stabilized by the attraction of the terminal nitro and methoxy groups, and (B,G) polar side-by-side association governed by group charges along the molecule, nitro-lateral methoxy attraction, and steric interactions of the lateral methoxys. (D,E) The NON-POL system exhibits distinct correlation functions for antiparallel and parallel molecular pairs, gNP anti (",z) and gNP par (",z). (D,H,I) The preferred antiparallel packing gives strong side-by-side correlations, governed by group charges along the molecule; and (D,J,K) weaker antipolar nitro-nitro end-toend association. (D,F,G) The parallel correlations in the NONPOL system are the most relevant to the stability of polar order in the NF phase, as they are determined by the inherent tendency of the molecular interactions for polar ordering in the presence of enforced polar disorder. Comparison of (B) and (D) shows identical preferred modes of parallel association in the two systems, with the POL system correlations being even stronger in the NONPOL system. This is clear evidence that the polar packing motifs giving the correlation functions (B) and (D), exemplified by the sample POL MD configurations (F) and (G), stabilize the polar order of the ferroelectric nematic phase. ! Intermolecular correlations -In order to make headway in understanding the roles of molecular structure and interaction in RM734 and its relation to polar ordering, we applied the atomistic simulations to probe molecular association and packing in the POL and NONPOL systems. We characterized molecular pair positional and orientational correlations by measuring several g(",z), the conditional densities of molecular centers about a molecule fixed with its center at the origin, with its long axis u in Fig. S22A parallel to z and pointing in the +z direction [nitro group near (",z) = (0,-10Å) and the methoxy group near (",z) = (0,10Å)]. Here the center of the molecule is defined as the midpoint of u in Fig. S22A, and the  g(",z) are angular averages about z of density over the azimuthal orientations of the molecule at the origin. The g(",z) are therefore independent of azimuthal angle ✦ , reflecting uniaxial nematic symmetry. They all exhibit a correlation hole (g(",z) ~0) around the origin and extended along z for ~ 1.5 molecular lengths where other molecular centers are excluded because of steric repulsion and the strong nematic orientational ordering. The pair distributions gP(",z) and gNP(",z) for the POL and NONPOL systems, shown in Fig. S22B, and S22C-E, respectively, display a number of striking features indicating specific molecular association motifs. In the POL system, pronounced arc-shaped peaks observed near (E 7 K#LM N 7 OPP#LF indicate a strong tendency for head-to-tail association of parallel pairs of molecules, as illustrated by the representative pair configuration shown in Fig. S22C. Such head-to-tail configurations are characterized by close association of positively charged H atoms in the terminal methoxy group of the molecule at the origin with negatively charged O atoms in the nitro group of its neighbor, suggesting that head-to-tail association is in large part the result of specific electrostatic interactions. Prominent off-axis peaks near E 7 Q#LM N 7 OR#L are also observed in gP(",z), and analysis of pair configurations associated with these peaks (e.g., Fig. S22E) reveals close association of oppositely charged atoms, including close contacts between positively charged H atoms in the lateral methoxy group and the negatively charged O atom in the terminal methoxy group, and between negatively charged O atoms in the nitro and carbonyl groups and positively charged H atoms in the phenyl rings. These observations also point to specific electrostatic interactions stabilizing polar pair configurations. In the POLAR simulation the lateral methoxys appear to be key to establishing the relative positioning of the side-by-side molecular associates that prefer polar ordering. The presence of a region of reduced probability ('correlation hole') in gP(",z) near E 7 Q#LM N 7 K#L shows that side-by-side configurations of parallel pairs of molecules are relatively unfavorable, presumably due to the excluded volume of their lateral methoxy groups.
The NONPOL system forces both antiparallel and parallel molecular pairs, which give correlation functions, gNP anti (",z) and gNP par (",z), that exhibit very strongly expressed, polarity-dependent molecular recognition. In the on-axis or nearly on-axis peaks of gNP anti (",z) in Fig. S22G we observe that the z # -z symmetry of gNP anti (",z) is the most strongly broken, as expected since HO-OH association will be different from OH-HO association. Thus, there is a prominent arc-shaped peak near E 7 S#LM N 7 ;TU#L, arising from HO-OH antiparallel lateral association of terminal nitro groups, illustrated by the representative pair configuration in Figs. S22G,K. Pair configurations associated with this peak are characterized by close contacts between negatively charged nitro O atoms and positively charged phenyl H atoms adjacent to the nitro group, suggesting that these configurations are electrostatically stabilized. In contrast, the OH-HO associations between terminal methoxy groups (associated with the peak near E 7 K#LM N 7 VPT#L) are very weak (Figs. S22G,J).
The NONPOL system contribution to gNP anti (",z) from side-by-side antiparallel pairs (Figs. S22G-I), shows a prominent peak near E 7 Q#LM N 7 ;U#L, associated with pair configurations characterized by close contacts between positively charged H atoms in the lateral methoxy group and negatively charged nitro O atoms (an example of which is shown in Fig. S2I). The side-by-side pair configurations associated with the peak near E 7 Q#LM N 7 ;P#L similarly involve close contacts between oppositely charged O and H at-oms in the ester groups and phenyl rings. These findings provide further support for the hypothesis that electrostatic interactions between specific oppositely charged atoms play a dominant role in stabilizing the characteristic pair configurations observed in our simulations.
The NONPOL system parallel pair correlation function gNP par (",z) in Fig. S22F is very similar to the POL system gP(",z) in Fig. S22C, indicating a nanosegregation of the par and anti components, a mixture of OH-OH-OH and HO-HO-HO chains with the OH-HO at their interfaces. Remarkably, the polar features of gP(E,z) are not only dominant in gNP par (",z) but even more pronounced than in gP(E,z) itself. This suggests that there are certain polar associations in the POL system that can reduce the overall polar order, but that can be replaced by antipolar associations in the NONPOL system that are more favorable for nearby polar order. In any case, the simulations show that the polar correlations in Figs. S22B,C,F,E, emergent in both POL and NONPOL systems, are the only packing motifs stabilizing polar order and therefore must be the principal drivers stabilizing the NF phase.
The PLUPOLAR Nematic -The POL simulation equilibrates a state in which end-to-end flipping is kinetically arrested and the periodic boundary conditions constrain the allowed wavelengths of orientation fluctuations to $x < 55Å and $z < 70Å. The remnant short ranged fluctuations create the pair correlations exhibited in Fig. S22, which are confined to the volume " <10Å and z < 30Å about the origin, molecular neighbor separation scales, well within the dimensions of the simulation box. These conditions create a PLUPOLAR (plus quam polar [38]) equilibrium state of constrained polar ordering yielding the simulated PS values in Fig. 3 (open circles). Comparing these values with the RM734 data shows that, on the one hand, in the PLUPOLAR state the fluctuations that lead to the phase transition are clearly suppressed, while the remnant short range fluctuations give a PS value exhibiting only a weak dependence on temperature. On the other hand, this PS gives a good account of the polarization density of the NF at low temperature, evidence that at low T the NF phase approaches some comparable PLUPOLAR-like condition of having only short-range fluctuations, and that the simulated g(",z) are characterizing their remnant correlations. This state may be glassy, if the strong T-dependence of the viscosity is any indicator.