Soliton walls paired by polar surface interactions in a ferroelectric nematic liquid crystal

Surface interactions are responsible for many properties of condensed matter, ranging from crystal faceting to the kinetics of phase transitions. Usually, these interactions are polar along the normal to the interface and apolar within the interface. Here we demonstrate that polar in-plane surface interactions of a ferroelectric nematic NF produce polar monodomains in micron-thin planar cells and stripes of an alternating electric polarization, separated by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${180}^{{{{{{\rm{o}}}}}}}$$\end{document}180o domain walls, in thicker slabs. The surface polarity binds together pairs of these walls, yielding a total polarization rotation by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${360}^{{{{{{\rm{o}}}}}}}$$\end{document}360o. The polar contribution to the total surface anchoring strength is on the order of 10%. The domain walls involve splay, bend, and twist of the polarization. The structure suggests that the splay elastic constant is larger than the bend modulus. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${360}^{{{{{{\rm{o}}}}}}}$$\end{document}360o pairs resemble domain walls in cosmology models with biased vacuums and ferromagnets in an external magnetic field.


Abstract.
Surface interactions are responsible for many properties of condensed matter, ranging from crystal faceting to the kinetics of phase transitions. Usually, these interactions are polar along the normal to the interface and apolar within the interface. Here we demonstrate that polar in-plane surface interactions of a ferroelectric nematic NF produce polar monodomains in micron-thin planar cells and stripes of an alternating electric polarization, separated by 180 o domain walls, in thicker slabs.
The surface polarity binds together pairs of these walls, yielding a total polarization rotation by Domains and domain walls (DWs) separating them are important concepts in many branches of physics, ranging from cosmology and high-energy science 1 to condensed matter 2-4 .
When the system cools down from a symmetric ("isotropic") state, it might transition into an ordered state divided into domains. For example, domains in solid ferroic materials such as ferromagnets and ferroelectrics exhibit aligned magnetic moments or electric polarization [2][3][4] .
Within each domain, the alignment is uniform, following some "easy direction" set by the crystal structure. These easy directions are nonpolar, thus opposite orientations of the polar order are of the same energy. The boundary between two uniform domains is a DW, within which the polar ordering either gradually disappears or realigns from one direction to another. By applying a magnetic or electric field, one can control the domains and DWs, which enables numerous applications of ferroics, ranging from computer memory to sensors and actuators [2][3][4] .
Recent synthesis and evaluation [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] of new mesogens with large molecular dipoles led to a demonstration of a fluid ferroelectric nematic liquid crystal (NF) with a uniaxial polar ordering of the molecules 13,14 . The ferroelectric nature of NF has been established by polarizing optical microscopy observations of domains with opposite orientations of the polarization density vector P and their response to a direct current (dc) electric field 13,14 . The surface orientation of is set by buffed polymer layers at glass substrates that sandwich the liquid crystal 13,14 . This sensitivity to the field polarity and in-plane surface polarity makes NF clearly different from its dielectrically anisotropic but apolar paraelectric nematic counterpart N.
In this work, we demonstrate that the surface polarity of in-plane molecular interactions produces stable polar monodomains in micron-thin slabs of NF and polydomains in thicker samples. The polar contribution to the in-plane surface anchoring potential is on the order of 10%.
The quasiperiodic polydomains feature paired domain walls (DWs) in which P realigns by 360 o .
The reorientation angle is twice as large as the one in 180 o DWs of the Bloch and Néel types that are ubiquitous in solid ferromagnets and ferroelectrics 2,3 and in a paraelectric nematic N 23 . The polar bias of the "easy direction" of surface alignment explains the doubled amplitude of the 360 o DWs and shapes them as coupled pairs of 180 o static solitons. The width of DWs, on the order of 10 μm, is much larger than the molecular length scale, which suggests that the space charge produced by splay of the polarization within the walls is screened by ions and that the splay modulus 1 in NF is significantly higher than the bend 3 counterpart. The enhancement of 1 is evidenced by the textures of conic-sections in NF films with a degenerate in-plane anchoring, in which the prevailing deformation is bend. Numerical analysis of the DW structure suggests that 1 / 3 >4 in the NF phase of the studied DIO material.

RESULTS
We explore a material abbreviated DIO 7  be contrasted to the textures in cells with antiparallel assembly, in which and ̂ twist along the normal -axis 13,14 .
where ≥ 0 and ≥ 0 are the apolar (quadrupolar, or nematic-like) and polar anchoring coefficients, respectively, Fig.1g. This form follows the one proposed by Chen et al. 14 and places a global minimum at = 0. When =0, the anchoring is polarity-insensitive, and the minima at = 0, ± are of an equal depth. As increases, the minima at = ± raise to the level Δ =      Therefore, ∥̂ and realignment of preserves the magnitude . This feature makes the observed DWs similar to Néel DWs in ferroics, as opposed to Ising DWs, in which → 0.
Reorientation of within each DW generates a "bound" space charge of density = −div . If the polarization charge is not screened by ionic charges, then the balance of the elastic energy (per unit area of the wall) and the electrostatic energy As envisioned by Meyer 34 and detailed theoretically in the subsequent studies [35][36][37] , the ionic screening enhances the splay elastic constant 1 associated with (div ̂) 2 in the Frank-Oseen free energy density: 1 = 1,0 (1 + 2 / 2 ), where 1,0 is the bare modulus, of the same order as the one normally measured in a conventional paraelectric N, and = √ 0 2 is the Debye screening length, which, for the typical parameters specified above and = 10 23 / 3 , is on the order of 10 nm. With ~10 nm, ~1 nm, the enhancement factor, 2 2~1 0 2 , could be strong.
Thus, 1 in NF can be much larger than 1 in N. Very little is known about the elastic constants in the N phase of ferroelectric materials and practically nothing is known about the elasticity of NF.
Chen et al 24 measured 1 ≈ 10 2 in the N phase of DIO and expected 37 1 ≈ 2 pN. Mertelj et. al. 10 reported that in the N phase of another ferroelectric material RM734, 1 is even lower, about 0.4 pN. Since the bend elastic constant 3 of NF is not supposed to experience an electrostatic renormalization, it is expected to be a few tens of pN; for example, Mertelj et. al. 10 found 3 ≈10-20 pN for the N phase of RM734. Therefore, the ratio = 1 / 3 in NF could be larger than 1, ranging from a single-digits value to ~10 2 . The next section presents qualitative evidence that 1 > Prevalence of bend in NF films with degenerate in-plane anchoring. The textures of N and NF are strikingly different when there is no in-plane anchoring. Figure 5 shows the textures of thin ( = 5 − 7 μm) films of DIO spread onto glycerin; the upper surface is free. Thermotropic N films are known to form 2 domain walls of the W type, stabilized by the hybrid zenithal anchoring, tangential at the glycerin substrate and tilted or homeotropic at the free surface 38 ; these 2 domain walls contain both splay and bend and are clearly distinguished in DIO as bands with four extinction bands, Fig.5a. The NF textures feature an optical retardance that is consistent with the director being tangential to the film. The most important feature is that the curvature lines of and ̂ are close to circles and circular arches, Fig. 5b,c, which implies prevalence of bend and signals that splay is energetically costly. One often observe disclinations of strength +1 with which satisfies the boundary condition ( , ) = 0 at = ± /2; is the tilt amplitude.
The Frank-Oseen free energy with the bulk, saddle-splay, and the azimuthal surface anchoring terms reads where 1 , 2 , 3 , and 24 are the elastic constants of splay, twist, bend, and saddle-splay, respectively. The equilibrium director field ̂|| minimizing the free energy in Eq. (4) where = √ 1 1+ , = 2arcsinh√ 1 ; "+" signs correspond to a = 1 pair in Fig. 2c,e and Supplementary Fig. 10a. The solution is a superposition of two -walls located at = ± 2 and limiting a stripe of a nearly uniform ↓↓ , Fig. 6a. The -soliton (7) The intensity of unpolarized monochromatic light, transmitted through two crossed polarizers enclosing a birefringent sample with a DW pair described by Eq. (7) and running parallel to one of the polarizers 27 , produces a texture with maximum light transmission at = /4, 3 /4, 5 /4, and 7 /4 and extinction at = 0, /2, , 3 /2, and 2 , Fig.6b, which is qualitatively similar to the experimental textures in Fig.4.
To facilitate a comparison with the experiment, the width of the DW pairs is characterized by distances /2 between the two central bright stripes, between two dark narrow stripes, 3 The first integral of the Euler-Lagrange equation is The -soliton solution corresponds to the particle rolling down the potential [ ] starting at = 0, where = 0, through the two wells, and arriving at = 2 . Because energy is conserved, the soliton would be stable as the maxima at = 0, 2 are both at = 0. To find ( ), one needs to impart a small initial "momentum" forcing the particle to start the motion. Figure 7 shows the results of numerical analysis. The width parameters /2 , , and 3 /2 of the DW pairs are not much affected by the elastic anisotropy when 1 / 3 ≪ 1, but increase, approximately as ∝ √ 1 / 3 , when 1 / 3 > 1, Fig. 7b. Because of their topological 2 -rotation nature, the DW pairs must incorporate both splay and bend, no matter the value of 1 / 3 . A notable qualitative feature of the director profile ( ) of the DW pairs is that as 1 / 3 increases, the stripes of splay widen, Fig. 7a. The structure tends to decrease the high splay energy by extending the length over which the splay develops; in contrast, it could afford a shorter bend development since 3 is low. Domain walls in a chiral smectic C (SmC*) stabilized by a magnetic field show similar features 40 , with the difference that, in SmC*, it is 3 that is increased by the ionic screening. Thus, it is the bend stripes that are wider in SmC* than their splay counterparts.
The effect of elastic anisotropy on the ratio 3 /2 / /2 is very strong when 1 / 3 is in the range 0.1-10, Fig. 7c. As 1 / 3 increases, the width of the splay region progressively expands and /2 approaches 3 /2 . When compared to the experimental value 3 /2 / /2 =1.8 obtained by averaging data of 64 DW pairs of both W and S types, the model of a planar -soliton suggests Equation (12)  Significantly thinner cells would hardly experience polar tilt at all: a strong zenithal anchoring (associated with the tilts away from the plane) makes the energetic costs of a vertical gradient over a short prohibitively high. Note, however, that our analysis is limited to a particularly a b c / / simple -dependence for both and and the quantitative estimates above might be changed by a more rigorous analysis.
To find the tilt configuration ( ), we minimize the Frank-Oseen free energy in Eq. (4) using gradient descent. The sharp bend of ( ) at large 1 / 3 introduces computational challenges. To get a qualitative picture while ensuring the numerical convergence of the gradient descent procedure, we take 1 / 3 = 10 and / 3 = 15, for which we expect a noticeable tilt. The resulting configurations of the polar angle ( ) and the tilt ( ) are shown in Fig. 8a,c,  ; c,d, two projected schemes of the polarization field in the one-quarter of the soliton in which we find the largest tilt , with the same parameters as in part (a). In all simulations, = 15 3 , 2 = 3 /2, and = 0.1.
The width ratio 3 /2 / /2 depends on the presence of tilt and the cell thickness, Fig.9b.
In thicker cells, the width ratio is smaller as the tilt allows for a faster reorientation of the azimuthal angle , as shown in Fig. 8a. The decrease, however, depends on the value of 1 / 3 , Fig. 9b. The dependence is subtle, with the width ratio approaching the planar value for small 1 / 3~( 1 − 4), but reaching a smaller value for 1 / 3 ≈ 10.  3) for various values of 1 / 3 and / 3 . Note the marked energy gain from introducing a tilt for thick cells. For thinner cells, / 3 < 10, the gain is negligible, especially at large ratios 1 / 3 . b, Ratio of width parameters 3 /2 / /2 vs 1 / 3 for different cell thicknesses / 3 . Note that this ratio is expected to be smaller whenever there is substantial tilt in the director configuration. For thinner cells, / 3 < 10, the ratio approaches the planar value (black line) for large 1 / 3 as the tilt becomes negligible. In all simulations, = 0.1 and 2 / 3 = 0.5. The dashed line shows 3 /2 / /2 =1.8 obtained by averaging experimental data for 64 DW pairs. The lines connecting the data points in these plots are a guide to the eye.
Comparison of experimental and numerical shapes of the domain walls. The width ratio 3 /2 / /2 can be used to estimate 1 / 3 , Fig. 9b. We analyzed the profiles of transmitted monochromatic light intensities similar to the one in Fig. 4c for DWs pairs in samples of thickness ranging from 4.6 μm to 15.9 μm, which implies 3 < / 3 < 6. In this range, there is no clear a b thickness dependence of the width ratio. The experimental data, averaged over 64 DWs pairs, yield 3 /2 / /2 = 1.8 ± 0.3. According to the model predictions in Fig.9b, the value 3 /2 / /2 = 1.8 corresponds to 1 / 3 = (4 − 7) in the model with polar tilts and / 3 = 10, and to 1 / 3 = 10 in the model of planar DWs. However, a relatively large standard deviation in the measured width parameter, ±0.3, embraces the possibility of much higher elastic anisotropy. An additional factor of uncertainty is in the strong dependence of the geometrical parameters and thus of 1 / 3 on the in-plane polar anchoring parameter , Fig. 7c. We thus conclude that the experiments on the structure of DW pairs place the lower bound on the elastic anisotropy of NF, 1 / 3 ≥ 4 , which is supported by both Fig.7c and 9b.

Discussion.
The In thin cells, could be large enough to eliminate the energy barrier between the = 0 and = states and cause the system to relax directly into the ground state ( ) = 0, see Supplementary Eq. S10 and Supplementary Fig. 13. In cells thicker than The geometry of the domains and DW pairs is defined primarily by the balance of the polar and apolar terms in the surface potential, suggesting potential applications as sensors and solvents capable of spatial separation of polar inclusions. The advantage of NF is that the material is fluid and is thus easy to process in various confinements. Since the domains form in an optically transparent and birefringent NF fluid with a high susceptibility to low electric fields, other potential applications might be in electro-optics, electrically-controlled optical memory, and grating devices.

Methods.
Sample preparation and characterization. The aligning agent PI-2555 and its solvent T9039, both purchased from HD MicroSystems are combined in a 1:9 ratio. Glass substrates with ITO electrodes are cleaned ultrasonically in distilled water and isopropyl alcohol, dried at 95 o C, cooled down to the room temperature and blown with nitrogen. An inert N2 environment is maintained inside the spin coater. Spin coating with the solution of the aligning agent is performed according to the following scheme: 1sec @ 500 rpm → 30 sec @ 1500rpm →1sec @ 50rpm. After the spin coating, the sample is baked at 95°C for 5 min, followed by 60 minutes baking at 275°C.
The spin coating produced the PI-2555 alignment layer of thickness 50 nm.
The PI-2555 layer is buffed unidirectionally using a Rayon YA-19-R rubbing cloth where we take the locations to be the midplanes of the thin slabs: = − /2 + ( − 1/2)Δ .
We then choose a large enough such that our matrix converges. Note that the intensity for crossed polarizers can be easily extracted from the matrix elements . We have the following expressions for the intensities when the polarizers are aligned along the and axes and when they are at 45 ∘ to these axes, respectively: + = | 12 | 2 and × = 1 4 | 11 + 21 − 12 − 22 | 2 .

Data availability
All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Competing interests
The authors declare no competing interests.           3) for various values of 1 / 3 and / 3 . Note the marked energy gain from introducing a tilt for thick cells. For thinner cells, / 3 < 10, the gain is negligible, especially at large ratios 1 / 3 . b, Ratio of width parameters 3 /2 / /2 vs 1 / 3 for different cell thicknesses / 3 . Note that this ratio is expected to be smaller whenever there is substantial tilt in the director configuration. For thinner cells, / 3 < 10, the ratio approaches the planar value (black line) for large 1 / 3 as the tilt becomes negligible. In all simulations, = 0.1 and with an eluent of hexane/EA: 10/1 to give two compounds. It turned out that the first spot is the target compound with trans-2,6-dioxane structure (0.32 g, yield 35%), and the second spot is the DIO isomer with cis-2,6-dioxane structure, as shown in Figure S1. The two isomers showed different peaks of the1,3-dioxane part in the NMR spectra. Figure 8. Temperature dependence of DIO birefringence; =535 nm, cell thickness = 6.8 μm.

III. Width of the domain
The width of the domains that corresponds to the distance between locations with = /2 and = 3 /2, shows a weak dependence on the cell thickness of planar cells.

IV. Soliton-soliton solution of Euler-Lagrange equation (3).
The solution satisfying the Euler-Lagrange equation (3) with the boundary conditions (±∞) = 0, (±∞) = 0 could also be written in a compact form: It is also interesting to study the soliton with → as → ±∞ accounting for the difference in the elastic constants and the polar tilt of the director. In the absence of tilt ( = 0), the soliton will decay to a constant = . However, introducing a tilt allows for the azimuthal angle to transition to = 0,2 where the anchoring is favorable. In this case, the soliton breaks up into two -solitons which move apart from each other, relaxing the system into a uniform = 0,2 . The motion of these walls for = 1 / 3 = 10, / 3 = 20, = 0.1, 2 / 3 = 0.5 is shown in Supplementary Fig. 12 where is the total free energy (Frank-Oseen and anchoring energy), with the ansatz ( , ) = ( ) sin(2 / ). We will set the relaxation coefficients to unity = = 1 for simplicity.
For our initial condition, we take the planar equilibrium configuration for the polar angle (black line in Fig. Z), and a nearly constant tilt ≈ 0.5 at the location of the soliton. The boundary conditions on our numerical solutions are = and = 0. After an initial transient, the tilt localizes at the center of the two -solitons, as shown in the red line in Supplementary Fig. 12.
Then, evolving the dynamics in Eq. (S4) pushes apart the -solitons, creating a region with = 0,2 , as shown with the solid lines in Supplementary Fig.12. The tilt amplitude forms two traveling bumps that move along with the -solitons, as shown with dashed lines in Supplementary   Fig. 12.

Supplementary Figure 12. Relaxation of a non-topological soliton via tilt.
Timeevolution of a non-topological soliton with = as → ±∞ which, in the case of no tilt, is shown with a solid black line (for = 10, / 3 = 20, = 0.1, 2 / 3 = 0.5). By introducing a tilt , the soliton [after an initial transient which sharpens the domain walls (red line)] splits into two regions with a substantial tilt (dashed lines) where rotates by . These two regions spread apart from each other due to the favorable anchoring energy for = 2 , as shown with the solid lines. We consider here simple relaxation dynamics, Eq. (S4), solved numerically.

VII. Energy of a twisted state.
A cell with a similar alignment of at the two plates carries no elastic energy. When (S10)