Domain patterns with paired soliton walls stabilized 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 produce stable domain structures in the bulk of a ferroelectric nematic liquid crystal. Monodomains form in micron-thick cells, while thicker cells feature quasiperiodic stripes of an alternating uniform electric polarization, separated by domain walls, within which the polarization rotates by


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 produce stable domain structures in the bulk of a ferroelectric nematic liquid crystal.
Monodomains form in micron-thick cells, while thicker cells feature quasiperiodic stripes of an alternating uniform electric polarization, separated by domain walls, within which the polarization rotates by 180 o .The surface polarity makes these walls paired with a total rotation by 360 o .
Analysis of the domain structures allows one to determine the polar contribution to the surface anchoring potential.The 360 o pairs of domain walls resemble domain walls in cosmology models with biased vacuums and ferromagnets in an external magnetic field.The polarity-biased ferroelectric structures are highly susceptible to weak electric fields and could lead to applications in advanced electro-optics, sorting of polar inclusions, sensing, memory, and grating devices.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][3][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] 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 nematic counterpart N.
In this work, we demonstrate that the surface polarity of in-plane molecular interactions produces stable monodomains in micron-thick slabs of NF and polydomains in thicker samples.
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 an apolar nematic N 21 .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 domain structures and their strong response to weak electric fields allow one to determine the polar contribution to the in-plane surface anchoring potential.

Results.
We explore a material abbreviated DIO 7 synthesized as described in the Supplementary Information, Figs.S1-S7.On cooling from the isotropic (I) phase, the phase sequence is I−174°C −N−82°C −SmZA−66°C −NF−34°C −Crystal, where SmZA is an antiferroelectric smectic 22 .The sandwich-type cells are bounded by two glass plates with layers of polyimide PI-2555 buffed unidirectionally.The plates are assembled in a "parallel" fashion, with the two buffing directions R at the opposite plates being parallel to each other.We use Cartesian coordinates in which R =(0,−1,0) is along the negative direction of the -axis in the  plane of the sample.The electric field is applied along the -axis.
Planar alignment.The N and SmZA phases show a uniform alignment of the optical axis (director  ̂) along the rubbing direction , Fig. 1a,b.In the absence of the electric field, depending on the cell thickness , NF forms either polydomain structures in thick samples,  > 3 μm, Fig. 1c, or polar monodomains in thin cells,  = 1 ÷ 2 μm, Fig. 1d.At the bounding plates,  and  ̂ are parallel to the surface, as evidenced by the measurement of optical retardance Γ = 250 nm of a cell with  = 1.35 μm, which yields the DIO birefringence Δ = Γ/=0.185,close to the previously reported value 22 .The planar alignment avoids a strong surface charge.Even a small tilt ~5 o of  from the  plane would produce a surface charge density   ~~4 × 10 −3 C m −2 , which is much larger than the typical surface charge (10 −4 ÷ 10 −5 ) C m −2 of adsorbed ions in nematics 23,24 ; here  ≈ 4.4 × 10 −2 C m −2 is the polarization of DIO 7 .
Ferroelectric monodomains in thin NF cells.Thin cells, filled in the N phase at 120°C, show a monodomain texture, with the polarization  = (0,1,0) antiparallel to  = (0, −1,0), 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     The DWs enclosing the narrow domains always exist and terminate in pairs.There are two types of the pairs.In the first, called W-pairs because of the shape of the director field, Fig. explained by the screening effect of ionic impurities 28 .The free energy per unit area of an NF cell, after integration over the cell thickness, writes Setting the variation of the energy to zero leads to the first integral of the Euler-Lagrange equation: For  = 0 and the boundary conditions In the presence of the electric field,  varies along both the and -axes, producing a complex combination of twist, bend, and splay, which is difficult to quantify.Qualitatively, the electric field antiparallel to  within a narrow domain surrounded by a W-pair of DWs could reorient  either CW or CCW because of the symmetry illustrated in Fig. 2c. Figure 2f shows an event when these two opposite directions of realignment happen within the same domain.In the S-case, only one direction of realignment is allowed, CCW in the case of Fig. 3a-c.

Discussion and Conclusion.
The polar nature of the azimuthal surface anchoring of NF brings about patterns of polar monodomains and polydomains with alternating directions of the polarization .DWs with 360 o rotation of the director could also be observed in a smectic C liquid crystal 28,30- 32 , in which case they are attributed to an externally applied electric field 26 or to the asymmetry of the film along the normal direction 32 .In a uniaxial apolar nematic N, 360 o DWs connect surface point-defects, called boojums, in a hybrid aligned film, in which one surface imposes a tangential orientation of  ̂ and another one sets a perpendicular alignment of  ̂, i.e, again the reason is the asymmetry with respect to the normal direction 33,34 .Under hybrid alignment of N, the 360 o DW carries an elastic energy ∝  proportional to their length  and width <<R, which is smaller than the elastic energy of an isolated boojum with an energy ∝  2 , where  is the characteristic size of the system 34 .Unlike all listed examples, the 360 o DWs in NF are caused by interactions that are polar in the plane of the bounding surfaces.The observed 360 o pairs of DWs are also different from 180 o DWs in NF cells with an antiparallel assembly of buffed plates that preset twist deformations 13,14,22 .The pronounced coupling between the surface polarity and the bulk structures allows us to estimate the polar contribution  =     = (0.03 − 0.13) to the in-plane anchoring of .
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.
The sandwich-type cells are assembled from glass plates with a spin-coated layer of polyimide PI-2555 (HD MicroSystems) of a thickness 50 nm.The PI-2555 layer is buffed unidirectionally using a Rayon YA-19-R rubbing cloth (Yoshikawa Chemical Company, Ltd, Japan) of a thickness 1.8 mm and filament density 280/mm 2 to achieve a homogeneous planar alignment.An aluminum brick of a length 25.5 cm, width 10.4 cm, height 1.8 cm and weight 1.3 kg, covered with the rubbing cloth, imposes a pressure 490 Pa at a substrate and is moved ten times with the speed 5 cm/s over the substrate; the rubbing length is about 1 m.Unidirectional rubbing of a polyimide-coated substrates is known to align a nematic in a planar fashion, with a small pretilt of the director  ̂.For example, the director of a conventional nematic 5CB in contact with a buffed PI-2555 makes an angle 3 °± 1 ° with the substrate; the tilt direction correlates with the direction R of buffing 35 .Two PI-2555-coated glass plates are assembled into experimental cells in "parallel" geometry, with the two buffing directions R at the opposite plates being parallel to each other.One plate contains a pair of parallel transparent indium tin oxide (ITO) stripe electrodes separated by 3-5 mm along the -direction.A Siglent SDG1032X waveform generator and an amplifier (Krohn-Hite corporation) are used to apply an in-plane dc electric field E=(0, ±1, 0).The optical textures are recorded using a polarizing optical microscope Nikon Optiphot-2 with a QImaging camera and Olympus BX51 with an Amscope camera.PolScope MicroImager (Hinds Instruments) is used to map the director patterns and measure the optical retardance.

Fig. 1d .
Fig.1d.A dc electric field  = (0,1,0) directed along  and of an amplitude  = (1 ÷ 10) kV/m causes no textural changes, while the opposite field polarity reorients  ̂ and  beginning with  ↓ = −1 kV/m, Fig.1d.As the field increases, the optical retardance Γ diminishes, Fig.1d, which indicates that  ̂ twists away from the rubbing direction in the bulk, Fig.1f.The decrease of Γ is caused by the formation of horizontal left-and right-twisted 180 o DWs of the Bloch type near the plates, Fig.1f.Above a critical field   = −11 kV/m, the surface anchoring that keeps  antiparallel to  ( ↑↓ ) is broken, and a uniformly aligned state  ↓↓  nucleates and propagates across the cell, swiping away the twisted state.Once formed, the  ↓↓  state is stable for days,

Figure 1 .
Figure 1.DOI textures in planar cells with parallel assembly.a, b, c, polarizing optical microscopy of a thick  =4.7 μm cell and d, e PolScope Microimager textures of a thin 1.35 μm cell; a, b, uniform N and SmZA textures, respectively; c, polydomain NF texture; in wide domains, the polarization  is antiparallel to the rubbing direction , in narrow domains,  is parallel to ; two 180 o DWs enclosing the narrow domain could reconnect (circles mark some reconnection points); d, field-induced realignment of  from the direction − to ; e, reversed field polarity realigns  back into the ground state  ↑↓ ; f, scheme of  reorientation in part d; there are two 180 o twist DWs of the Bloch type near the plates; g, azimuthal surface anchoring potential for different ratios of the polar   and apolar   coefficients.

3 ,
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 = 2  and become local, until disappearing at   ≥   , Fig.1g.The energy barrier   =   (1 + ) 2 /2 at  = arccos (−) separates the global and local minima;  =   /  is the relative strength of the polar anchoring.The surface anchoring torques 25 ()  | =0, = (  sincos +   sin)| =0, resist the realigning action of the field, Fig.1f.For a small deviation from the preferred state  = 0, the torque is   +   ; for a deviation from the metastable state  = ± the torque is weaker,   −   .These torques compete with the elastic torque  2 /  = √ 2  caused by the field-induced twist of  in subsurface regions of a characteristic thickness   = √ 2 / , where  2 is the twist elastic constant, Fig.1f.The difference in the surface torques explains the difference in the reorienting fields, which allows one to determine the relative strength of the polar anchoring,  =   /  ≈ 0.13.The measured  ↑ = 0.6 kV/m, | ↓ | = 1 kV/m, reported  = 4.4 × 10 −2 C/m 2 7 , and a reasonable assumption 25  2 ≈ 5 pN, lead to the estimates   ≈ 0.3 μm,   ≈ 1.3 × 10 −5 J/m 2 , and   ≈ 1.7 × 10 −6 J/m 2 .The estimated   is within the range reported for nematics at rubbed polyimides 26,27 .Note here that in the thin cell under study, the material was filled by a capillary flow along the − direction, Fig.1d.Filling a cell by a flow along  yields  ↓ = −1.4kV/m and  ↑ = 1 kV/m, which implies a weaker polar bias:  ≈ 0.08.This flow effect on the surface anchoring deserves further study, but to describe the polydomain patterns in thick cells, we avoid it by filling the cells in the isotropic state and then rapidly cooling the sample to N with a rate 30°C /min, followed by slow cooling to NF with the rate 2°C/min.Ferroelectric domains in thick NF cells.Cooling a cell with  = 4.7 µm from the SmZA phase results in a quasiperiodic domain texture of NF, with alternating wide (width   =5÷150 μm) and narrow (  =1÷ 2 μm) domains, in which  ↑↓  and  ↓↓ , respectively, Fig.1c, as established by the response to the electric field, Figs.2,3.The coexistence of the  ↑↓  and  ↓↓  domains results from the two-minima surface potential ().Both narrow and wide domains are extinct when aligned along the polarizer or analyzer of a polarizing optical microscope, which confirms that  ̂ and  are collinear with .Within the wide domains, Γ =900 nm and Δ =0.19, i.e.,  ̂ and  are in the  plane.Once formed, the domains remain stable for days.Repeating heating-cooling cycles, even following a crystallization or melting into the isotropic phase, reproduces the same qualitative NF patterns.

Figure 2 .
Figure 2. Topologically stable 360 o W-pairs of DWs.a, textures observed between crossed polarizers with  ↓↓  in the narrow central domain separated by two bright 180 o DWs from the wide domains with  ↑↓  at the periphery; the electric field realigns  in the narrow or wide domains, depending on the field polarity; b, the same textures, observed with an optical compensator that allows one to establish the reorientation direction of ; c, structure of the 360 o W-pair of DWs with  = 1; d, field-driven realignment of  into two opposite directions within the same 360 o W-pair of DWs; e, azimuthal angle variation   (/;  = 0) across a single  DW of the Néel type, Eq.(4), and across a  soliton-soliton pairs   (/;  > 0), Eq.(5), for various 's shown in the legend.

2 ,
Domains of opposite polarization are separated by DWs.Within each DW,  ̂ and  realign by 180 o , as established by optical microscopy observation with crossed polarizers and an optical compensator (λ -plate, 532 nm).The DWs form pairs that enclose narrow domains and either run from one end of the sample to the other or reconnect with each other, Fig.1c.The elastic energy density stored within the wall, where  is the average Frank elastic constant,    is the Boltzmann's energy, ~1 nm is the molecular size, and   ≈ (5 ± 2) μm is the width of each DW, Fig.2c, is much lower than the energy density     3 of the orientational order.Therefore,  ∥  ̂ and realignment of  over   preserves the magnitude .This feature makes the observed DWs similar to Néel DWs in ferroic materials, as opposed to Ising DWs, in which  → 0.

Figure 3 .
Figure 3. Topologically trivial 360 o S-pair of DWs.a, geometry and b, texture of CW and CCW rotations of , no field; c, geometry and d, e, f, textures of realignment by the electric field that erases the narrow S-domain at  = 3.3 kV/m; g, h, the electric field of an opposite polarity tilts  in two wide domains, but (i) does not cause a complete reorientation, contrary to the case of the narrow domain in f.
2c,  rotates by 180 o in the same fashion in both DWs, either clockwise (CW) or counterclockwise (CCW).In the second type, met less frequently and called 360 o S-pairs for their geometry, Fig.3a, the rotation directions alternate: if  rotates CW by 180 o in one DW, it rotates CCW by 180 o in the next one.The difference between the W-and S-pairs is topological, as illustrated by mappings of the oriented line  threaded through the DWs pair and the enclosed domain, into the order parameter space, a circle  1 25 , Figs.2c, 3a,c.Each point on  1 corresponds to a certain .The line  in Fig.2c produces a CCW-oriented closed contour Υ encircling  1 once.The W-pair of CCW walls thus carries a topological charge  =1 25 .A DW pair with a CW 360 o rotation of  would carry  = −1.Neither could be transformed into a uniform state  =0 without breaking the surface anchoring and overcoming a large elastic energy barrier.S-pairs of 180 o -walls with alternating sense of rotations are topologically trivial,  =0: the corresponding contour Υ does not encircle  1 fully and could be contracted into a single point  = 0 without the need to overcome the elastic energy barrier, Fig.3c.To elucidate the structure of DW pairs, consider the balance of elasticity and surface anchoring, assuming that  varies only along the  -axis.Electrostatic effects caused by gradients of , flexoelectricity, order electricity, and ions are neglected.A partial justification is that the studies of DWs in ferroelectric smectic C demonstrated the absence of electrostatic contributions,

2 𝐾 2 ( 1 −𝜔 2 )𝑑( 1 −
Fig.S10.In cells thicker than   ≈  2  2 8  ≈ 3.6 μm,   is smaller than the energy 4  of the metastable uniform state () = .In these thick cells, the local energy minimum at  =  and the energy barrier that separates  =  and  = 0 directions are preserved (Supplementary Fig.S10); thus the system could relax into either the ground state, () = 0, or the metastable state () = , which explains the observed domain structures with DWs in the thick samples.We limited our analysis by the structures observed in the deep NF phase, but the experiments show rich dynamics of the emerging patterns during cooling in the high-temperature end of the NF phase, most likely caused by the temperature dependencies of   ,   , and the elastic constants; these will be described elsewhere.