## Abstract

Although the existence of the twist-bend (N_{TB}) and splay-bend (N_{SB}) nematic phases was predicted long ago, only the former has as yet been observed experimentally, whereas the latter remains elusive. This is especially disappointing because the N_{SB} nematic is promising for applications in electro-optic devices. By applying an electric field to a planar cell filled with the compound CB7CB, we have found an N_{TB}-N_{SB} phase transition using birefringence measurements. This field-induced transition to the biaxial N_{SB} occurred, although the field was applied along the symmetry axis of the macroscopically uniaxial N_{TB}. Therefore, this transition is a counterintuitive example of breaking of the macroscopic uniaxial symmetry. We show by theoretical modeling that the transition cannot be explained without considering explicitly the biaxiality of both phases at the microscopic scale. This strongly suggests that molecular biaxiality should be a key factor favoring the stability of the N_{SB} phase.

## INTRODUCTION

The nematic (N) phase is the most common thermotropic liquid crystalline state. In the usual nematic, the rod-like achiral molecules orient on average parallel to a common direction, **n**, called the “director.” The nematic long-range orientational order is described by a uniaxial second-rank tensor, *Q*_{ik} = (*S*/2)(3*n*_{i}*n*_{k} − δ_{ik}), with the scalar order parameter, *S* = ⟨3cos^{2}β − 1⟩/2, where β is the angle between the molecular long axis and the director, and the brackets denote a statistical average (*1*). Any deviation of *S* from its equilibrium value, *S*_{0}(*T*), costs a huge energy. In contrast, the nematic energy is unaffected by a uniform rotation of the director and only the director distortions, imposed by the boundary conditions or by external fields, cost some elastic energy (*2*)**s** = **n**(∇ · **n**), **b** = **n** × (∇ × **n**), and the pseudoscalar *t* = − **n** · (∇ × **n**) describe the main distortion modes splay, bend, and twist, respectively. At equilibrium, the N phase is uniform, with ∇**n=** 0. However, the chiral nematic (N*) phase, or cholesteric, formed by chiral molecules, is spontaneously twisted, with **n** following a right-angle helix of pitch, *p**, usually larger than a few hundred nanometers. Then, the twist term in Eq. 1 becomes *K*_{22} (*t* − *t*_{0})^{2}, where *t*_{0} = ± 1/*p** is the spontaneous twist induced by the molecular chirality.

Different nematic phases are expected when the mesogenic molecules are less symmetric. For example, a biaxial nematic has been predicted (*3*) when the molecular structure and interactions are strongly biaxial. However, although it was reported in lyotropic systems, it is most elusive in thermotropic ones. For strongly bent mesogenic molecules, studies based on different physical mechanisms predicted the existence of a nematic phase with spontaneous bend distortion of the director (*4*, *5*). However, unlike twist, a pure bend distortion cannot fill space without introducing energy-expensive defects. Mixed-distortion modes (*4*, *5*), either twist + bend or splay + bend, avoid this constraint and lead to two modulated nematic (MN) phases, the twist-bend (N_{TB}) and splay-bend (N_{SB}) phases, respectively. In the N_{TB} phase, the director is arranged on a heliconical structure (Fig. 1), **n** = (sinθsinφ, sinθcosφ, cosθ), where 0 < θ < π/2 is the heliconical tilt angle (i.e., the angle between **n** and the helix axis, **h**), φ = *qZ* is the phase of the precession of **n** around **h**, and *q* = ± 2π/*p* is the wave number of the director “wave” of wavelength *p*. The sign of *q* describes the handedness of the helix. Even with achiral molecules, the N_{TB} phase is chiral, with broken chiral symmetry (*5*) and doubly degenerate chirality, and it consists of right and left monochiral domains. The occurrence of the N_{TB} phase then appears as an especially interesting case of spontaneous chiral symmetry breaking.

In contrast, the N_{SB} phase is achiral when the molecules are achiral. It can be regarded, approximately, as a sine-wave oscillation of the director (Fig. 1) **n** = (0, sin (θ_{0}sinφ), cos (θ_{0}sinφ)), where θ_{0} < π/2 is the oscillation amplitude and φ ≈ *qZ* is its phase. Therefore, the N_{SB} phase is described by a linearly polarized wave of director distortion. By symmetry, all the orientations of the polarization are equivalent and correspond to the same continuously degenerate ground state. Note that this phase should not be confused with the “splay nematic” phase that was recently discovered with strongly polar calamitic mesogens (*6*–*8*). The splay nematic phase is characterized by a spontaneous splay distortion instead of the spontaneous bend distortion that induces the N_{TB} and N_{SB} phases.

The three different nematic phases (N, N_{TB}, and N_{SB}) expected for bent-shaped mesogenic molecules have different macroscopic symmetries. For achiral molecules (considered here), the N phase is uniaxial and achiral. The N_{TB} phase, however, has broken chiral symmetry and is locally biaxial. Because of the strong spontaneous bend distortion, the uniaxial symmetry around **n** is broken. However, because of the precession of **n** around **h**, the N_{TB} phase remains macroscopically uniaxial (at scales much larger than the pitch *p*). The N_{SB} phase is achiral, like the N phase, but is locally biaxial, like the N_{TB} phase. Moreover, the N_{SB} phase is biaxial even at a macroscopic scale because the plane in which **n** oscillates is unique (Fig. 1).

In both MN phases, the bend is spontaneous, but the splay and twist cost some elastic energy. Therefore, the relative stability of the two phases depends on *K*_{11}/*K*_{22}, and the stable phase is N_{TB} for *K*_{11}/*K*_{22} ≥ 2 and N_{SB} otherwise (*5*, *9*).

Starting from 2008, several groups reported a second, low-temperature, nematic phase, called N_{X} (originally, the mnemonic N_{X} was used for a nematic phase with an unknown structure; subsequently, it has been taken to denote the N_{TB} phase) for bent-shaped mesogens, which presented some of the properties predicted for modulated nematics (*10*–*13*). Then, Cestari *et al.* (*14*) demonstrated that the N_{X} phase is the predicted N_{TB} phase. Later studies confirmed this assignment (*15*) and showed either indirectly (*16*, *17*) or in direct FFTEM (freeze-fracture transmission electron microscopy) (*18*, *19*) and resonance x-ray scattering (*20*) experiments that *p* is extremely small (*p* < 10 nm). Most of the defining properties of the N_{TB} phase were also confirmed: the temperature-dependent conical angle (*16*, *18*, *19*, *21*), the chiral symmetry breaking (*14*, *22*), the existence of large monochiral domains (*16*, *18*, *21*), and the strong pretransitional decrease of *K*_{33} in the nematic phase (*14*, *23*–*25*).

Further synthetic work (*26*–*29*) and experimental studies demonstrated that the N_{TB} phase is quite widespread for bent-shaped dimers. In stark contrast, the N_{SB} phase has yet to be observed, in the absence of an external constraint, for any compound. The elusiveness of the N_{SB} phase indicates that the low *K*_{11}/*K*_{22} ratio is only a qualitative criterion for its thermodynamic stability. However, an N_{TB} to N_{SB} transition could be induced by the topological constraint of a defect wall between two monochiral N_{TB} domains with opposite chiralities (*21*).

Strong fields applied to the N_{TB} phase may also induce phase transitions. Under a magnetic field of 25 T, the bent-shaped CB7CB [1″,7″-*bis*(4-cyanobiphenyl-4′-yl) heptane] dimer showed a weak decrease (∆*T*_{NTB − N} = 0.7 K) of the N-N_{TB} transition temperature (*30*). A similar effect is also expected (*31*, *32*) when a strong electric field, **E**, is applied parallel to **h** in the N_{TB} phase of a compound with positive dielectric anisotropy, ∆ε > 0. So far, the estimates of the field strength required for a measurable transition shift range from a few (*31*, *33*) to 100 V/μm (*32*). For ∆ε < 0 and/or **E** ⊥ **h**, the uniaxial symmetry of the N_{TB} phase is broken and the N_{TB} cone becomes elliptical, leading eventually, as expected, to an N_{TB} to N_{SB} transition (*31*–*33*). Strong planar surface anchoring may also act as a field, breaking the uniaxial macroscopic symmetry of the N_{TB} phase, leading to an elliptical N_{TB} cone (*34*) and, in the case of extremely strong anchoring, to a surface-induced transition to the N_{SB} phase (*35*).

Here, we report on a field-induced N_{TB}-N_{SB} phase transition, of CB7CB, with ∆ε > 0, in a quite different geometry. In a cell with planar alignment of the helix axis, **h**, we apply a strong, in-plane electric field, **E** ∥ **h**. At *E* = 0, we observe the usual N-N_{TB} phase transition at temperature *T*_{NTB − N}. Under a field, we observe a small decrease of *T*_{NTB − N}, as expected. However, the birefringence Δ*n*(*T*) measured in the low-temperature phase differs drastically from those of both the N and N_{TB} phases. Instead, it is similar to that measured (*21*) in a defect wall with an N_{SB} structure. This unexpected result indicates a symmetry-breaking transition from the macroscopically uniaxial N_{TB} phase to a field-induced macroscopically biaxial N_{SB} phase. This transition then appears as a clear example of uniaxial symmetry breaking. Using a simple model, we show that the field-induced transition is due to the biaxial order at the microscopic scale in both N_{TB} and N_{SB} phases.

## RESULTS

### Polarized optical microscopy

The field-induced phase transition is observed in a planar cell filled with the dimer CB7CB (see Fig. 2 and Materials and Methods for details) with a field applied parallel to **n** (in the nematic phase) or to **h** (in the N_{TB} phase). In the nematic phase (*T* = 103.4°C), without a field (Fig. 3A), the alignment of the sample is excellent, in the active region (AR) between the electrodes and above the electrodes, with **n** parallel to the substrates and to the rubbing direction **r**. Under the electric field (Fig. 3B, *E*= 8 V/μm), the director orientation remains the same in the AR, except close to the electrode edges. The field distribution in the cell (Fig. 2B) is highly uniform, with **E** ∥ **r**, in the middle of the AR. Moreover, no birefringence variation is observed under the field, which indicates that *S* is not affected by the field. In contrast, the birefringence varies rapidly above the electrode edges because **n** lies parallel to the tilted field there (Fig. 2B).

Upon slow cooling at *E* = 0, the nematic-isotropic (N-I) transition occurs at *T*_{N − I} = 115.8 ± 0.2°C, in reasonable agreement with previous studies (*14*, *16*, *29*). A small biphasic region of 0.4°C was measured at the center of the field of view, and we also observed a weak temperature variation, of about 0.2°C, along the AR. The texture of the N phase remains uniform, with **n** ∥ **r** (fig. S1). The N-N_{TB} phase transition occurs at *T*_{NTB − N} = 103.1 ± 0.1°C by progressive growth of domains (fig. S1, A to C), and after the last nematic domains have disappeared, the texture is again approximately uniform (fig. S1D), with the N_{TB} optic axis **N** ∥ **r**. Upon further cooling, the typical textural instabilities (*14*) of the N_{TB} phase appear (fig. S1, C to J): stripe texture, rope texture, and finally focal conic domains. These textural variations are essentially reversible, albeit with some hysteresis far away from *T*_{NTB − N} (fig. S1, K and L).

Upon slow cooling under the field (8 V/μm), the N-I transition occurs at *T*_{N − I} = 115.5 ± 0.3°C, revealing a moderate heating by the field in the AR. Throughout the temperature range of the N phase, the texture remains as in Fig. 3B but with the interference color varying with the temperature. The N-N_{TB} transition is first observed above the electrodes, at *T*_{NTB − N} = 103.1°C. At 103.0°C, this whole area is in the N_{TB} phase (Fig. 3, C and D), with homeotropic alignment close to the electrode edges because of the strongly tilted field there. However, in the AR, the sample still remains in the nematic phase, suggesting a small decrease of *T*_{NTB − N} under the field, by Δ*T*_{NTB − N} = 0.1° to 0.2°C, as expected by analogy with the magnetic field case (*30*). This shift of *T*_{NTB − N} might also be due to the Joule heating of CB7CB by the electric field, estimated to ~0.1°C in our in-plane field geometry.

Upon further slow cooling, the region over the electrodes remains in the N_{TB} phase, and its texture evolves through defect propagation from homeotropic to smectic-like focal conic defects (Fig. 3, E to L). Meanwhile, in the AR, a phase transition to another nematic-like phase occurs at 102.9°C. Because both phases are well aligned by the field and have almost the same birefringence, the transition is hardly visible under crossed polarizers (Fig. 3E). However, with a Berek compensator in the optical path (Fig. 3F), the sharp limit between the two phases is easily detected because they differ slightly in their birefringence. Because of a small in-plane temperature gradient across the cell, the new phase first appears in the cooler left-hand bottom corner of the sample and spreads slowly toward the hotter (by 0.1°C) right-hand top corner. The phase-separation line has a parabolic meniscus-like shape consistent with a field-induced shift of the transition temperature as Δ*T*_{NTB − N} increases with increasing field strength (the field being stronger close to the electrode edges).

At first glance, the textural observations of the low-temperature phase are compatible with its expected N_{TB} nature. However, detailed measurements (see the next section) revealed that its birefringence is drastically different from those of both the nematic and the N_{TB} phases, indicating the distinct nature of the field-induced phase. Therefore, for the time being, we will label this yet to be identified, low-temperature, nematic phase as “N_{Y}” (we avoid the ambiguous notation N_{X} because it was already used in a different context).

Upon further cooling in the field, the AR remains in the N_{Y} phase, but its texture progressively evolves. Close to the N_{Y}-N transition, the texture remains perfectly uniform, with **N** ∥ **r** (Fig. 3, G and H), which allowed us to measure precisely the birefringence. When the temperature is decreased further, some stripes appear in the AR (Fig. 3, I and J), with a contrast much weaker than in the absence of the field (for comparison, see fig. S1). Farther from the N_{Y}-N transition, the stripes become more marked but are still hardly visible between the crossed polarizers without a compensator. In this temperature range, the birefringence measurements are still possible with a reasonable precision. Upon further cooling, these stripes have greater contrast (Fig. 3, K and L) and some focal conic domains appear close to the electrode edges, preventing any precise birefringence measurements.

Close to *T*_{NY − N}, the texture changes induced by temperature variations at constant field are reversible, with little or no hysteresis. Moreover, at a fixed temperature in this range, a fast birefringence response is observed upon variation of the field strength (fig. S2). Both observations suggest that the N_{Y} phase is not a smectic but a modulated nematic phase despite its smectic-like textures at lower temperatures. Such textures are expected not only from the N_{TB} phase but also from other MN phases (such as the N_{SB} phase), because of their pseudo-layered structures (*14*, *32*, *36*).

Further from the transition, the response of the texture to a variation in temperature shows more hysteresis (about 2° to 3°C at 88°C) and the response to a variation in field strength completely disappears. Switching the field on and off at 88°C has practically no effect on the texture, which is apparently “frozen” because of the stiffening of the pseudo-layered structure, the entanglement of the defects, and the smectic- and N_{TB}-like increase of the viscosity.

To summarize the influence of temperature and electric field intensity, because the birefringence values measured under field are not influenced by the thermal history, the existence of the field-induced N_{SB} phase does not depend on the thermodynamic path used experimentally.

### Birefringence of the field-induced nematic phase

Qualitatively, the behavior of the N_{Y} phase is that expected for the N_{TB} phase subject to a strong field, **E**, oriented parallel to the optic axis **N** (*31*, *32*). Macroscopically, the torque applied by the field to the local N_{TB} director **n** is elastically transmitted to **N**, which favors the uniform alignment of **h** parallel to **E**. Close to *T*_{NTB − N}, the torque is large enough to ensure uniform alignment of **N**. At lower temperatures, the torque is too weak to affect the stiffer macroscopic elasticity of the N_{TB}-phase and the intrinsic textural instabilities of the phase reappear (Fig. 3, I to L).

Apart from the macroscopic alignment, sufficiently strong fields could modify the microscopic heliconical structure by decreasing the cone angle, leading eventually to a second-order N_{TB}-N transition (*31*, *32*). Experimentally, this should result in a tiny, field-dependent decrease of the N_{TB}-N transition temperature, as observed here. However, the detailed test of this hypothesis is difficult in the electric field case because of a possible local heating of the sample. On the contrary, when the texture is uniformly aligned, the precise measurement of the birefringence, Δ*n*, provides the field dependence of the conical angle and, more generally, information on the microscopic structure of the phase (*21*). In the AR, where Δ*n* is measured, the texture is perfectly uniform in the nematic phase, with or without field. However, it is only sufficiently uniform over ~0.2°C below the transition in the N_{TB} phase (without field), and over ~5° to 8°C below the transition in the N_{Y}-phase (for fields between 4 and 20 V/μm).

The temperature dependence of Δ*n* was measured under the field in the N and N_{Y} phases (Fig. 4), and a few data points were also measured in the N phase at *E* = 0. These points cannot be distinguished from those measured under the field, which confirm that the field is too weak to affect *S*. For comparison, Fig. 4 also shows previous birefringence measurements in the N and N_{TB} phases of CB7CB at *E* = 0 in a very thin (1.6 μm) cell (*21*). This cell displayed, in the N_{TB} phase, large and perfectly uniform monochiral domains, with right-handed (R) and left-handed (L) chirality. The birefringence is the same in the R and L domains, and it is drastically different from the Haller extrapolation of Δ*n* from the nematic state, because of the increase of the conical angle of the N_{TB} helix (*16*) from ~10° to ~35° upon decreasing temperature (*21*). A completely different birefringence has been measured in the defect walls separating the R and L domains (*21*). This revealed that the structure in the walls of the modulated nematic is N_{SB} instead of N_{TB} because of the topological incompatibility of the helices of opposite chiralities. Both N and N_{SB} “cores” of the defect wall would satisfy the topological constraint, but the N_{SB} structure costs less energy (*21*) than the nematic one because it keeps the spontaneous bend distortion (at least partially).

Throughout the nematic phase, the birefringence curves, measured with or without the field, are almost superimposed (Fig. 4). In stark contrast, the Δ*n* data measured in the N_{Y} phase under the field are completely different from the previous zero-field N_{TB} data (*21*) but lie very close to the data previously measured in the N_{SB} defect wall. This provides clear evidence that a large enough field **E** parallel to **N** induces an N_{TB}-N_{SB} transition, instead of the N_{TB}-N transition expected theoretically. Therefore, the N_{Y} phase is a field-induced N_{SB} phase.

### Theoretical model

To understand the unexpected field-induced N_{TB}-N_{SB} transition, we compare the energies of the N, N_{TB}, and N_{SB} phases as a function of the temperature and the applied field. In the absence of a field, two models have been proposed to obtain the N-N_{TB}-N_{SB} phase diagram. The first that we will follow here is the “elastic-instability” model (*5*). It considers the MN phases of bent-shaped molecules as usual uniaxial nematics with a negative *K*_{33}. The elastic energy is given by *f*_{el} = *f*_{II} + *f*_{IV}, where *f*_{II}, the Frank elastic energy (Eq. 1), is quadratic in **n** gradients, whereas *f*_{IV}, which is quartic in **n** gradients, is required to bind the energy from below when *K*_{33} < 0. As the bend elastic constant changes sign at the N-N_{TB} transition, it is expected that close to the transition temperature, *T*_{NTB-N}, *K*_{33} depends linearly on the temperature: *K*_{33}(*T*) = *a*(*T* − *T**), and the weak temperature dependence of all the other elastic constants can be neglected (*23*, *25*). Here, *T** is the temperature of the second-order N-N_{TB} phase transition predicted by the model. (Because the N-N_{TB} transition for CB7CB is weakly first order, as explained in Discussion, *T** is slightly different from the experimentally observed transition temperature, *T*_{NTB-N}). The experimental results for *K*_{33}(*T*) in the nematic phase of CB7CB (see fig. S3) confirm the predicted linear dependence and provide the slope, *a* = 0.066 pN/K.

To model the electric field effects in the N and MN phases, we also need to couple the nematic order parameter tensor, **Q**, with the electric field. Only the quadratic (in **E**) dielectric coupling is important here because the polar flexoelectric contribution to the total energy integrates to zero (*16*, *32*). Therefore, the electric contribution to the free energy density is Δ*f*(*E*) = − ε_{0}**E** · ε · **E**/2, where ε is the dielectric susceptibility tensor. In general, like **Q**, ε has different symmetry and orientation with respect to **E** in the three considered phases. The average conformation of the bent-shaped dimer only has C_{2v} symmetry (Fig. 5A) and is therefore polar and biaxial. We choose the molecular frame **123** with the **3**-axis parallel to the long molecular axis and the **2**-axis along the *C*_{2} axis. The N phase has a higher symmetry, D_{∞h}, and we choose the director-frame **xyz** with the *z* axis along the *D*_{∞} axis (i.e., along **n**). Then, the orientational order is described by the usual uniaxial tensor, **Q*** ^{u}*, which is diagonal in the

**xyz**frame. In the MN phases, however, the macroscopic uniaxial symmetry is broken by the strong spontaneous bend,

**b**. There is no longer revolution symmetry around

**n**and the phases are biaxial and polar (

*5*,

*9*,

*36*,

*37*), at least locally, at scales smaller than

*p*. Therefore,

**Q**is biaxial,

**Q**=

**Q**

*, and choosing the*

^{b}**axis parallel to**

*y***b**(Fig. 5A), we have in the director frame (

*38*)

*S*is the usual uniaxial scalar order parameter and

*P*, the secondary order parameter, describes the biaxiality of the MN phase. In principle,

*S*and

*P*can be calculated using molecular field theory and atomistic modeling (

*14*,

*39*–

*41*). However, close to the N-MN transition, a simple symmetry-based approach gives

*P*= β

_{2}

**b**

^{2}≪

*S*(see Materials and Methods), where β

_{2}is a material constant.

The dielectric tensor, diagonal in the director frame, is uniaxial in the N phase^{u}_{xx} = ε^{u}_{yy}) and it is biaxial in the MN phases**I** is the unit tensor and ^{b} are material constants (see Materials and Methods). Although ε^{b} is the same in the two MN phases in this approximation, the dielectric contribution to the energy is different because of the different orientations of the director frame with respect to the field (Fig. 5).

In the absence of the field, the minimized (i.e., ground-state) energy densities of the N, N_{TB}, and N_{SB} phases are respectively [see (*5*) and the Supplementary Materials]*C* is a fourth-order elastic modulus, and for the N_{SB} case, we used the “harmonic” approximation, θ_{sb}(*Z*) = θ_{0} sin (*q*_{sb}*Z*).

In this approximation, the N-N_{TB}-N_{SB} phase diagram as a function of the dimensionless ratios *R*_{3} = *K*_{33}(*T*)/*K*_{22}, and *R*_{1} = *K*_{11}/*K*_{22} is very simple (Fig. 6A). For *R*_{3} > 0, i.e., for *T* > *T**, the only solution is the N phase, with *R*_{3} < 0, the energies of the modulated phases are negative, resulting in a second-order N-MN transition at the line *T* = *T**). [The predicted transition is second order due to the approximations made in (*5*), but our experiments and more recent theoretical models (*42*) show that the N_{TB}-N transition is first order.] The transition line between the two modulated phases occurs at *et al.* (*9*) in a flexoelectric model for the N-MN transition. The analytical approximation also gives *9*).

Subject to the field, we expect a field-induced shift, *i* = 1 or 3, of the phase boundary lines. Let us first neglect the biaxiality of the MN phases and assume that ε remains uniaxial, ε = ε^{u}, as in the N phase. Because **E** is parallel to the **Z** axis of the laboratory frame (Fig. 5, B and C), the electric term reads

As in the absence of the field, we obtain after energy minimization [see (*32*) and the Supplementary Materials]

From these expressions, we obtain two field-induced N-N_{TB} and N-N_{SB} transitions, with the same field threshold, *T*_{NTB − N} = Δ*T*_{NSB − N}. At both of these N-MN field-induced transitions, the amplitude of the modulation, θ or θ_{0}, respectively, is continuous (see fig. S4). Note that the modulation amplitude is a natural order parameter of the transition (it is finite in the “ordered” phase and zero in the “disordered” phase), and its continuity indicates that the transition is second order. In contrast, the wave number, *q*, jumps at the transition from its large finite value in the MN phase to an undefined value in the N phase (in the absence of a modulation wave, the wave number has no physical meaning).

The N-N_{TB}-N_{SB} phase diagram for uniaxial coupling with the field (Fig. 6B) is very similar to that obtained at *E* = 0. The N_{TB}-N_{SB} transition still occurs at _{TB}-N_{SB} transition cannot occur when the dielectric tensor is uniaxial. The only field effect on the phase diagram is the shift of the MN-N transition by _{TB}-N transition, which are already known from previous studies. With *E* = 8 V/μm, θ_{tb} ≈ 0.2 rad (*21*), *16*, *18*, *19*), Δε^{u} ≈ 2 (*25*, *43*), *K*_{22} ≈ 8 pN (*25*), and *a* ≈ 0.066 pN/K [estimated by extrapolating the linear part of *K*_{33}(*T*) in the N phase, see fig. S3], we obtain Δ*T**(*E*) ≈ 0.7°C. This value is larger than the observed shift, ≤0.2°C, which shows that the model overestimates the effect of the field on the transition.

If we now consider the biaxiality of ε, proceeding as before, we obtain (see the Supplementary Materials)*r*(*E*) is again given by Eq. 7 and_{2}δε^{b} > 0), i.e., the biaxial dielectric coupling favors the N_{SB} phase with respect to the N_{TB} one.

The phase diagram for the biaxial case is shown in Fig. 6C. Although the position of the N-MN transition line remains exactly the same as in the uniaxial case, the N_{TB}-N_{SB} transition now occurs at

Last, Fig. 6D shows the phase diagram at fixed temperature, *T* < *T**, as a function of the field strength and *R*_{1} ratio. The N phase is stable above the threshold field _{SB} for _{TB} otherwise.

## DISCUSSION AND CONCLUSIONS

Qualitatively, the biaxial coupling with the electric field provides a satisfactory explanation of the experimentally observed features: the small decrease of the N-MN transition temperature and the field-induced N_{TB} to N_{SB} phase transition. This emphasizes the important role of the biaxial order in both modulated nematic phases. In the N_{TB} phase, the biaxial order is only “local,” at a scale smaller than the pitch, as the macroscopic symmetry of the phase is uniaxial (*9*, *16*, *34*, *36*, *37*). Therefore, for properties, like the birefringence, that are defined at the macroscopic scale, the local biaxiality plays only a minor role (*21*). In contrast, for the field-induced N_{TB} to N_{SB} phase transition, the biaxial coupling with the field acts on the microscopic structure of the phase and is not macroscopically averaged. Moreover, the dielectric biaxiality of the MN phases in the director frame, ε^{b}_{yy} − ε^{b}_{xx} = β_{2}δε^{b}**b**^{2}, is important because of the large values of β_{2} and δε^{b}. β_{2} is large because of the efficient coupling between the bent shape of the dimer and the director bend, leading to the spontaneous bend, and δε^{b} is large because of the much larger polarizability of the molecule in the **yz** plane than along the **x** axis. Furthermore, although **ε** is almost the same in both MN phases in the director frame, its macroscopic average component along the field, ⟨ε^{b}_{ZZ}⟩, differs in the two phases because of the different orientations of **Q**^{b} with respect to the field.

This combination of geometrical structure and local biaxiality of the two MN phases explains our most marked and counterintuitive result: An electric field, **E**, applied parallel to the symmetry axis **N** of the macroscopically uniaxial N_{TB} phase, induces a transition to the macroscopically biaxial N_{SB} phase. This effect is closely related to the local biaxial symmetry of the nematic order in both N_{TB} and N_{SB} phases because of the alignment of the biaxial bent-shaped molecule in the strongly bent director field. If this biaxiality is neglected, then no N_{TB}-N_{SB} phase transition is expected when **E** ∥ **N**. In contrast, the transition was previously predicted and also reported in a completely different geometry where the field was applied perpendicular to the macroscopic symmetry axis, **E** ⊥ **N** (*31*–*33*, *44*). However, the phase transition is then intuitively expected because the uniaxial symmetry of the system is actually broken by the field. Similarly, the usual surface anchoring anisotropy, with the zenithal anchoring much stronger than the azimuthal, also breaks the uniaxial macroscopic symmetry of the N_{TB} phase and has been predicted to promote a transition to the N_{SB} phase (*35*).

Quantitatively, though, the experimental behavior is not properly captured by the elastic instability (*5*) and the flexoelectric (*9*) theoretical models because they both predict values of _{TB}-N_{SB} phase transition in zero field. Because the typical *K*_{11}/*K*_{22} ratio is between 1 and 2 for nematics, including bent-shaped mesogens, the N_{SB} phase should be much more common than the N_{TB} one. In stark contrast, no experimental observation of the N_{SB} phase was reported so far, whereas the N_{TB} phase has been observed in numerous compounds.

This discrepancy might be due to an approximation adopted in both models. The modulated nematic is considered as an extension of the usual nematic, with *S* continuous at the transition (*S*_{N} = *S*_{MN}) and uniform (∇*S*_{MN} = 0), and with periodic modulation of **n**. This approximation may be adequate for the N_{TB} phase due to its roto-translational symmetry (**Q** is unchanged by an arbitrary translation ∆*Z* along **h** and simultaneous rotation by ∆φ = ∆Z/*p* around **h**, leading to ∇*S*_{NTB} = 0 and uniform amplitude of the bend). However, in the N_{SB} phase, the bend is not uniform but instead oscillates along the modulation axis, and ∇*S*_{NSB} ≠ 0. Then, *S* should also oscillate, in phase with **b**^{2}(*Z*), *S*_{NSB}(*Z*) = *S*_{N} + δ*S*_{NSB}**b**^{2}(*Z*), resulting in an additional energy and shifting the N_{TB}-N_{SB} transition line to a lower value of *K*_{11}/*K*_{22}.

For CB7CB, we have observed a symmetry-breaking N_{TB}-N_{SB} transition under a moderate electric field. The low-field threshold of the induced phase transition comes from the large biaxial anisotropy of the molecule, δε^{b}, and the strong coupling between **b** and *P* (*P* = β_{2}**b**^{2}) in the two modulated nematic phases. Moreover, the low threshold indicates that, in the absence of a field, the representative point of the N_{TB} phase of CB7CB in the generic phase diagram must lie very close to the N_{TB}-N_{SB} transition line. A small renormalization of *K*_{11} and *K*_{22} by the field is enough to induce the transition. All these ideas provide important clues for guiding future efforts toward the synthesis of compounds presenting the elusive N_{SB} phase in the absence of a field. Considering the important role of biaxiality in the N_{TB}-N_{SB} energy balance, strongly biaxial molecules (e.g., dimers or bent-core molecules with highly biaxial rigid cores, parallel to the **23**-plane in the molecular frame of reference) are expected to promote the N_{SB} phase.

Because of the well-known equivalence (*1*, *45*) between electric and magnetic field effects in liquid crystals, the N_{TB}-N_{SB} transition can also be expected when a strong magnetic field is applied to the N_{TB} phase of a dimer with strongly biaxial magnetic susceptibility tensor. A possible indication of this effect is the birefringence behavior (*30*) observed in CB7CB under a magnetic field *B* of 25 T. In the field, the value of Δ*n* is higher and its temperature dependence, Δ*n*(*T*), is very small compared to the steeper Δ*n*(*T*) curve in the zero-field case [see figure 3A in (*30*)]. This behavior is similar to that observed in our electric field experiment and might be due [at least partially, because in (*30*) the stripe texture of the cell also influences Δ*n*(*T*)] to a magnetic field–induced, uniaxial symmetry-breaking N_{TB}-N_{SB} transition.

Our observation of an electric field–induced transition from the N_{TB} to N_{SB} nematic phase demonstrates the important role of the local biaxiality of the MN phases. From a macroscopic point of view, this transition is counterintuitive: An electric field applied parallel to the symmetry axis of the macroscopically uniaxial N_{TB} phase induces a transition to the macroscopically biaxial N_{SB} phase. Our simple model taking into account the local biaxial order in the modulated phases describes qualitatively this transition. However, further development of the model, based on the biaxial order and its coupling with the nematic order parameter, is needed to reach a more quantitative description of the N-N_{TB}-N_{SB} phase diagram. Progress in this direction is required to help guide the search for the still elusive N_{SB} phase.

The discovery of N_{SB}-forming compounds will be an important breakthrough in liquid crystal applications. Because of the biaxiality of the N_{SB} phase (at all scales, from the molecular to the macroscopic ones), novel electro-optic effects are expected even in the usual liquid crystal cell geometry, i.e., in planar alignment with the field applied perpendicular to the plates. These effects would lead to low-threshold voltages and on and off response times substantially faster than those of the usual nematics. The N_{SB} phase will also present important advantages compared to the smectic phases: easy surface alignment, good uniformity and mechanical stability of the textures, and self-healing of the defects that may appear under mechanical, thermal, or electric shocks.

## MATERIALS AND METHODS

### Liquid crystal material

The mesogen used in the present study is the classic N_{TB}-forming, bent-shaped, liquid crystal dimer, CB7CB. There are two main reasons for this choice: The first one is that CB7CB is the best-characterized N_{TB} material and most of its physical properties and their temperature dependences are reported in the literature, which greatly helps the interpretation of our results. The second reason is that CB7CB is particularly suitable for optic and electro-optic studies of the N_{TB} phase: It has relatively large dielectric anisotropy, Δε ~ 2, which gives rise to a strong response to the applied field; it is stable, without any degradation or shift of the phase-transition temperatures even when submitted for several days to strong electric fields, *E* > 10 V/μm; and, last, it forms, in thin cells, large monochiral domains in the N_{TB} phase (*16*, *21*), allowing precise measurement of the birefringence in these domains and also in the defect walls between them.

The CB7CB had been carefully prepared using the methods described by Cestari *et al.* (*14*) and Barnes *et al.* (*46*), while Jokisaari *et al.* (*47*) give more recent, subtle details. The identity of the low-temperature nematic phase is given by Cestari *et al.* (*14*) as an N_{TB} nematic, with the transition temperatures, *T*_{NTB-N} and *T*_{N-I}, found to be 376 K and 389 K, respectively. These phase transitions were found to be first order in keeping with the N_{TB} and N phases having the same symmetries (*42*); the respective entropies of transition, ∆S/R, are 0.34 and 0.31, respectively (*46*).

### Electro-optic cell and experiment

For our study, we need to apply a strong and uniform electric field parallel to the macroscopic symmetry axis, **N**, of an N_{TB} single domain. In principle, a natural choice for this kind of experiment would be to use a thin sandwich cell, with surface alignment layers providing homeotropic alignment of the liquid crystal. In this geometry, an electric field, **E**, applied to the nematic phase using transparent electrodes on the inner surfaces of the cell would be parallel to the surface-imposed director orientation, **E** || **n**. In the N_{TB} phase, the homeotropic surface alignment of the director, **n**, would favor the homeotropic alignment of the helix axis, resulting in **E** || **N**, as desired. Unfortunately, this simple and attractive geometry of the experiment cannot be used in our case because it remains impossible to align CB7CB homeotropically (as for almost all the other N_{TB}-forming bent-shaped dimers). Actually, all the usual surface treatments used to achieve homeotropic alignment of nematics give, in the N and N_{TB} phases of CB7CB, a strong and inhomogeneous planar alignment of **n** and **N**, respectively. Even when a strong field (*E* = 10 V/μm) is applied, only the bulk of the sample is aligned homeotropically, while the texture remains markedly inhomogeneous in the vicinity of the surfaces.

To resolve this issue in our electro-optic experiment, we used a cell with homogeneous planar alignment and applied a strong in-plane electric field parallel to the macroscopic symmetry axis of a well-aligned N_{TB} single domain. Commercial liquid crystal cells do not match these requirements. In the most common type of cells, the field is applied across the cell gap, i.e., perpendicular to the alignment direction of the nematic, thus breaking the macroscopic uniaxial symmetry of the phase. Even in-plane–switching (IPS) cells, in which an in-plane electric field is applied by using interdigitated electrodes on one of the plates, are not suitable for our purposes. In these cells, the field is very inhomogeneous, because of the small pitch of the electrode comb, and it is perpendicular to the alignment direction of the nematic so that the macroscopic uniaxial symmetry of the phase is again broken.

To overcome this issue, we prepared homemade cells with in-plane field applied parallel to the planar alignment of the director, **n**, in the nematic phase, and of the helix axis, **h**, in the N_{TB} phase. The cell architecture is presented in Fig. 2A. The top substrate is a glass plate without electrodes; the bottom one is a glass plate with a low-resistivity transparent indium-tin oxide (ITO)–deposited layer. This layer was etched to form two parallel electrodes on the plate, separated by an inter-electrode distance, *L*_{e}, of 50 μm. The cell gap, *d*, of ≈6 μm, is defined by ball spacers placed at the corners of the substrates, i.e., far away from the inter-electrode region. Figure 2B shows schematically the electric field lines when a voltage *U* is applied across the electrodes. The electric field **E** is perpendicular to the electrode edges and, midway between the electrodes, it is approximately uniform and parallel to the substrates, with an amplitude *E* ≈ *U*/*L*_{e}. On both sides of this region and close to the electrode borders, the direction and strength of the field vary rapidly because of edge effects.

Both substrates were covered with a cured and uniaxially rubbed polyvinyl alcohol (PVA) film to impose uniform, planar alignment of the nematic director parallel to the rubbing direction, **r**, which was set perpendicular to the electrode stripes. Therefore, in the nematic phase, the electric field in the region of interest is uniform and parallel to the director (**E** || **r**). In the N_{TB} phase, the axis of the director cone, **N**, and the helix axis, **h**, are both expected (*34*) to orient parallel to **r**, and therefore to **E**, as required for our experiment.

After sealing the cells with a photopolymerizable glue (NOA 81, Norland), the liquid crystal was filled by capillarity into the cells at 120°C in the isotropic phase and then cooled to the nematic and the N_{TB} phases for the birefringence measurements. The temperature of the cell, placed on a heating stage (HS400, Instec), was controlled with 30-mK accuracy using a temperature controller (STC200D, Instec). All of the observations made with polarizing optical microscopy were performed with two microscopes: a Leitz 12 PolS microscope equipped with a Sony XC-HR70 grayscale digital camera and a color Nikon D50 camera and a Leica 2500P. The birefringence measurements (see Fig. 2C) were performed on the stage of the Leica 2500P with a Berek tilting compensator and a highly sensitive photodetector system. The latter consists of a photomultiplier tube (PMT; H6780, Hamamatsu) and an optical system, which allows the precise measurement of the intensity of the transmitted light in a small rectangular window in the image plane. We typically used a 15 μm × 60 μm window with its long side parallel to the electrodes and placed in the middle of the 50-μm inter-electrode gap, where the applied field is uniform and parallel to the optic axis of the sample, **n** or **N,** in the nematic and N_{TB} phase, respectively.

A first approximate measurement of the birefringence was made with the usual Berek compensator technique. The slow axes of the cell and of the compensator were oriented perpendicular to each other and at 45° with respect to the crossed polarizers. Then, the tilt angle of the compensator plate, 2*i*, was varied to achieve the best compensation of the phase shifts introduced by the cell and the compensator, providing the value of the optical retardation of the cell, *d*Δ*n*. Although this classical technique is very direct and fast, it is not precise enough for our purposes, because of the typical error bars of 10 nm on *d*Δ*n*. This low precision originates from the low sensitivity of the human eye in detecting the minimum of the transmitted light intensity at the compensation. To avoid this issue, we measured the transmitted intensity from the voltage drop of the PMT anode current on a large load resistance (1 megohm), which was actually the input resistance of a digital oscilloscope (DPO3014, Tektronix) used to average the signal using up to 512 acquisitions. In this way, we measured the transmitted intensity as a function of the compensator tilt angle, *I*(2*i*) (see Fig. 2D), and the parabolic fit of this function provides very precisely its minimum, which decreases the error bars on the optical retardation to <0.5 nm.

The signal applied to the electrodes was typically a square or sinusoidal alternating current (AC) waveform with a frequency in the range of 10 to 40 kHz. This signal was produced by an arbitrary waveform generator (TGA12101, TTI) with an amplitude *U* ≤ 10 V and was then amplified by a wide-band amplifier (Krohn-Hite 7402 M) up to an amplitude of *U* ≤ 400 V. This was the maximum voltage applied with a square waveform, resulting in a root mean square (RMS) in-plane field in the cell, *E*_{RMS} ≈ 8 V/μm. When needed, higher RMS voltages, with a sinusoidal waveform, were applied with a homemade transformer, which further amplified the Krohn-Hite output voltage up to *U*_{RMS} ≈ 1000 V. In most of our experiments, we used a cell, with *d* = 6.3 μm, with an applied square waveform AC field of *E*_{RMS} = 8 V/μm.

### Dielectric susceptibility tensor

The biaxiality of the modulated nematic phases is described by the secondary order parameter *P*. In principle, it is possible to calculate the order parameters *S* and *P* from molecular field models (*14*, *39*–*41*). However, in practice, this task is very difficult for bent-shaped dimers like CB7CB, and here, we will apply a simpler approximate approach based on the symmetry of the phases. Taking into account the molecular structure of the CB7CB dimer, we expect that *P* > 0, i.e., that the dimer is preferentially oriented with its **2**-axis parallel to the ** y** axis of the director frame. This optimizes the orientation of the highly anisotropic rigid parts of the two monomer units with respect to the curved director field. To a first approximation, in the vicinity of the N-MN phase transition, we can assume that

*S*is continuous at the transition and neglect its weak temperature dependence (

*21*). In contrast,

*P*= 0 in the uniform nematic phase but has a finite value in the MN phases, resulting from the large spontaneous bend

**b**. We can then consider

*P*as a scalar function of the scalar

**b**

^{2}and develop it in a series,

_{2i}are constants, whose temperature dependence is supposed to be weak and will be neglected. Taking into account that

**b**= 0 and

*P*= 0 in the N phase, we have β

_{0}= 0. Moreover, close to the transition,

**b**

^{2}can be considered as a small parameter, and we can truncate the expansion, retaining only the first term,

*P*= β

_{2}

**b**

^{2}. This simple expression is expected to apply to both the N

_{TB}and N

_{SB}phases because their local structures, at a scale much smaller than the pitch, are similar (a nematic phase with strong spontaneous bend distortion). In that sense, these two phases can be considered in a first approximation as two different “macroscopic textures” of the same spontaneously modulated nematic phase.

In both the uniaxial and biaxial cases, the dielectric tensor is diagonal in the director frame and can be expressed as a function of the order parameter tensor, which is, respectively, uniaxial, **Q**^{u}, or biaxial, **Q**^{b}**I** is the unit tensor and ^{u}, and δε^{b} are constants describing, respectively, the isotropic part of ε and its uniaxial and biaxial anisotropy. These constants are related to the components of the dielectric tensor in the molecular frame, ε^{M}

Taking into account that the dielectric tensor of the N or MN phase is a statistical average of the molecular tensor ε^{M}, ε = 〈ε^{M}〉, and considering the particular cases of a completely disordered (isotropic, *S* = 0, *P* = 0) and perfectly ordered uniaxial (nematic, *S* = 1, *P* = 0) or biaxial (modulated nematic, *S* = 1, *P* = 1) phases, we obtain

We note that ^{u}, and *S* can be measured in the nematic phase, just above the N-MN transition, which reduces the unknown parameters to δε^{b} and *P*, which are related to the biaxiality of the bent-shaped molecule and the modulated nematic phase, respectively. Moreover, we expect that the polarizability of the bent-shaped dimer is much larger in the plane **23**, because of the highly anisotropic rigid cores, than along axis **1**. Because the main contribution to ε^{M} is due to the molecular polarizability, we expect ε^{M}_{22} > ε^{M}_{11}, i.e., δε^{b} > 0.

During the review process of our manuscript, Tomczyk and Longa (*48*) published a theoretical model that considers the role of the biaxiality of the order parameter tensor in modulated nematic phases. Using a generalized mean-field model, they predicted two distinct N_{TB} nematic phases, N_{TB} (macroscopically uniaxial) and N_{TB,B} (macroscopically biaxial), for bent-core molecules. Further development of the model of (*48*) to describe the phase behavior under applied fields may clarify further the complex phase behavior of the modulated nematic phases and their broken uniaxial symmetry.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/6/36/eabb8212/DC1

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We are indebted to T. Lubensky for discussions.

**Funding:**This work was supported by the Agence Nationale pour la Recherche ANR (France) through grant BESTNEMATICS, ANR-15-CE24-0012, and by the Université de Picardie Jules Verne, Amiens, France.

**Author contributions:**C.B. and I.D. designed the in-plane field cells and performed the birefringence measurements as a function of temperature and applied field. C.M., G.R.L., and I.D. developed the theoretical model. P.D., I.D., and G.R.L. wrote the manuscript with an input from all coauthors.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or in the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2020 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).