Solid ferroelectrics are polydomain. Within each domain, the spontaneous electric polarization 𝐏𝐏 aligns along a certain rectilinear crystallographic axis (1)(2)(3)(4). As first proposed by Landau and Lifshitz (5), the domains form in response to a finite size of samples in order to reduce depolarization fields. Domains with a differently oriented 𝐏𝐏 are separated by domain walls (DWs), which are generally flat, as dictated by crystallographic axes and crystal facets (1)(2)(3)(4)(5)(6).
The recently discovered ferroelectric nematic liquid crystal (N F ) (7-10) is a liquid with a macroscopic spontaneous polarization 𝐏𝐏. This polarization is locally parallel to the director 𝐧𝐧 � ≡ -𝐧𝐧 � , which specifies the average quadrupolar molecular orientation (11). The polarization direction could be aligned by confining the material between two glass plates with rubbed polymer coatings (9,10,(12)(13)(14)(15)(16)(17)(18)(19)(20). In these samples, the DW shape is defined by the anisotropic surface interactions with the "easy axis" of the substrate and by the orientational elasticity of N F .
The DWs in a surface-aligned N F are rectilinear (13,19), zig-zag (12,(16)(17)(18) , lens-like (10,18), or smoothly curved (9,10,12,15,17,18,20). In samples with air bubbles, one observes incomplete parabolic walls which separate concentric patterns of the polarization imposed by the air-nematic interface and a uniform domain set by a rectilinear easy axis at the substrate (21). Experiments with fully degenerate in-plane surface anchoring (22) reveal that the prevailing type of domains not constrained by crystallographic axes and azimuthal anchoring are domains with (a) nearly uniform polarization or (b) nearly circular polarization, which implies bend deformation of 𝐏𝐏.
Splay of 𝐏𝐏 is diminished because it creates a bound ("space") charge of bulk density 𝜌𝜌 𝑏𝑏 = -div 𝐏𝐏 and thus increases the electrostatic energy. The avoidance of polarization splay in defect textures has been described previously for ferroelectric smectics C by Link et al. (23). In the azimuthally degenerate N F films, DWs separating a circular and a uniform domain are parabolas (eccentricity 𝑒𝑒 = 1), while DWs between two circular domains are hyperbolas (𝑒𝑒 > 1) (22). The eccentricity can vary along the DW; as a rule, 𝑒𝑒 < 1 near the tip of the DW. The observed domain textures minimize the bound electric charge of bulk density 𝜌𝜌 𝑏𝑏 = -div 𝐏𝐏 and of the surface density 𝜎𝜎 𝑏𝑏 = (𝐏𝐏 1 -𝐏𝐏 2 ) • 𝐤𝐤 ̂ at the DWs separating two neighboring polarization patterns 𝐏𝐏 1 and 𝐏𝐏 2 . Here 𝐤𝐤 ̂ is the unit normal to a DW, pointing towards domain 1 (22,24). To reduce 𝜎𝜎 𝑏𝑏 , a DW must bisect the angle between 𝐏𝐏 1 and 𝐏𝐏 2 , so that 𝐏𝐏 1 • 𝐤𝐤 ̂= 𝐏𝐏 2 • 𝐤𝐤 ̂ , making the projection of polarization onto 𝐤𝐤 ̂ continuous across the DW, while the tangential component changes sign. The remarkable bisecting properties of conics have been elucidated millennia ago by Apollonius of Perga (25). However, the bound charges are still present when the polarization realigns continuously in the plane of the sample along 𝐤𝐤 ̂ over a finite DW width. The projection 𝐏𝐏 • 𝐤𝐤 ̂= 𝑃𝑃 𝑘𝑘 of the polarization onto 𝐤𝐤 ̂ yields a non-vanishing bound charge density -𝜕𝜕𝑃𝑃 𝑘𝑘 /𝜕𝜕𝜕𝜕. This produces two oppositely charged sheets at the DW (21,26,27). The situation is reminiscent of electrically charged Néel walls in solid ferroelectrics, in which the polarization realigns in a plane perpendicular to the wall (28). The presence of some bound charge in the N F samples is also evidenced by observations of 2𝜋𝜋 soliton DWs with splay-bend realignment of 𝐏𝐏 being topologically protected (19) and of DWs separating circular domains with opposite sense of polarization circulation (22).
A full description of polarization patterns with the bound charges should involve the analysis of the (partially screened by ions) electrostatic energy (29), which is a difficult task. One often uses a simplified model, in which the electrostatics is reduced to a renormalization of the splay elastic constant (30)(31)(32)(33), 𝐾𝐾 1 = 𝐾𝐾 1,0 (1 + 𝜆𝜆 𝐷𝐷 2 /𝜉𝜉 𝑃𝑃 2 ) . Here 𝐾𝐾 1,0 ~10 pN is the bare splay modulus of the same order as the one measured in a conventional paraelectric nematic (N),
is the Debye screening length, 𝜉𝜉 𝑃𝑃 = � 𝜀𝜀𝜀𝜀 0 𝐾𝐾 1,0 𝑃𝑃 2 is the polarization penetration length, 𝜀𝜀 0 is the electric constant, 𝜀𝜀 is the dielectric permittivity of the material, 𝑒𝑒 = 1.6 × 10 -19 C is the elementary charge, 𝑛𝑛 is the concentration of ions, 𝜕𝜕 𝐵𝐵 is the Boltzmann constant, and 𝑇𝑇 is the absolute temperature. For the typical 𝑛𝑛 = 10 23 m -3 , 𝜀𝜀 = 10 -100 (34,35), 𝑃𝑃 = 6 × 10 -2 C/m 2 (10), and at room temperature, one finds 𝜆𝜆 𝐷𝐷 ≈ (10 -30) nm and 𝜉𝜉 𝑃𝑃 ≈ (1 -2) nm. Note that here we do not use the often-reported exaggerated values of 𝜀𝜀 since these represent an artifact of dielectric measurements in N F cells (34,36). Since 𝜆𝜆 𝐷𝐷 > 𝜉𝜉 𝑃𝑃 (10), 𝐾𝐾 1 in the N F should be much larger than 𝐾𝐾 1,0 in the N and larger than the twist 𝐾𝐾 2 and bend 𝐾𝐾 3 elastic constants in the N F .
Experimental data on elastic properties of N F -forming materials are scarce.
Chen et al. (18) measured 𝐾𝐾 1 ≈ 10𝐾𝐾 2 in the N phase of DIO and expected 𝐾𝐾 1 ≈ 2 pN (33). Mertelj et. al. (37) reported that in the N phase of the ferroelectric material RM734, 𝐾𝐾 1 is even lower, about 0.4 pN. Since the bend elastic constant 𝐾𝐾 3 of N F does not experience an electrostatic renormalization, it is expected to be a few tens of pN; Mertelj et. al. (37) found 𝐾𝐾 3 ≈10-20 pN for the N phase of RM734. Therefore, the ratio 𝐾𝐾 1 /𝐾𝐾 3 in the N F could be larger than 1, ranging from single-digits to ~10 2 . Studies of 2𝜋𝜋 DWs in N F (19) suggest 𝐾𝐾 1 /𝐾𝐾 3 > 4. The goal of this work is to explore whether a model with a strong disparity of the elastic constants, 𝐾𝐾 1 ≫ 𝐾𝐾 2 , 𝐾𝐾 3 , can explain the experimentally observed polarization patterns with DWs in the shape of conic sections. In Section 2, we present the experimental optical microscopy textures of these patterns. In Section 3, we propose a model which employs the well-known director-based Oseen-Frank energy with splay, bend, twist, and saddle-splay terms
for a nematic liquid crystal occupying a region Ω . However, since in the materials under consideration, the polarization vector tends to align with the director 𝐧𝐧 �, we replace 𝐧𝐧 � with 𝐏𝐏 when modeling these ferroelectric nematic textures, and ignore associated orientability issues that may arise. This Oseen-Frank type energy for the polarization is supplemented by a potential term that serves to set the preferred value for the magnitude of polarization, and by an anchoring term that strongly favors tangential anchoring on the surface of the sample, thus penalizing the component of 𝐏𝐏 normal to the surface. Most crucially, we pursue an asymptotic regime where the cost of splay is dominant over the other terms in the elastic energy density, that is 𝐾𝐾 1 ≫ 𝐾𝐾 2 , 𝐾𝐾 3 . To simplify the model further, we ignore saddle splay by setting 𝐾𝐾 2 + 𝐾𝐾 4 = 0.
We explore two N F materials, abbreviated DIO (8) and RM734 (7). On cooling from the isotropic (I) phase, the phase sequence of DIO, synthesized as described previously (19), is
where SmZ A is an antiferroelectric smectic (18). RM734 of purity better than 99% is purchased from Instec, Inc. The material is additionally purified by silica gel chromatography and recrystallization in ethanol. Its phase sequence is I-188°C -N-133°C -N F -84°C -Crystal. The N F samples are of three types: (ii) Flat cells of RM734 are assembled from glass plates spin-coated with thin (50 nm) layers of polystyrene, separated by a distance h = (1-10) μm and sealed with an epoxy glue Norland Optical Adhesive (NOA) 65. Polystyrene aligns 𝐏𝐏 tangentially (22,38). Silica spheres are added to some samples, Fig.
(iii) Freely suspended DIO films formed in square openings of metallic grids used as holders for samples in transmission electron microscopy. A 5wt% solution of DIO in dodecane is heated to 90 °C and then spread across the openings. After solvent evaporation, freely suspended DIO films form.
Below we analyze the shape of the DWs and the type of deformations they carry.
To analyze the textures, we use both the conventional polarizing optical microscopy and PolScope approach, invented by Oldenburg (39,40), applications of which to liquid crystals has been described in Refs. (41,42). Briefly, a PolScope represents a polarizing optical microscope with a variable optical compensator(s), which might be a nematic liquid crystal cell controlled by an electric field. The image of a sample is recorded in polarized light multiple times with different settings of the optical compensator; the numerical analysis of the set of transmitted light intensity maps reconstructs the two-dimensional maps of the optical axis (in projection onto the plane of imaging) and optical retardance. The approach assumes that the optic axis does not change along the light propagation direction. The PolScope observations in this study are performed by the Exicor Microimager (Hinds Instruments) operating at four wavelengths, 475 nm, 535 nm, 615 nm, 655 nm, which allows one to characterize samples with optical retardance up to 3500 nm.
The textures in samples (i) and (ii) show that the colloidal spheres trigger circular domains of polarization and DWs of parabolic shape separating such a circular domain from a domain with a nearly uniform polarization, Fig. 1. In particular, Figure 1(a) shows the in-plane map of the optic axis, which is parallel to the director and to 𝐏𝐏 in the studied materials. The DW shapes are fitted with an equation of a conic, written in polar coordinates (𝑟𝑟, 𝜓𝜓) with the origin at the core of a circular vortex,
where e is the eccentricity, and d is the distance from the core to the directrix. The fitted values are listed in Figs. 1, 2, and 3; the accuracy is better than 5%. The eccentricity of DWs separating a circular vortex and a uniform domain is close to 1, hence the name "P-wall", where "P" stands for the "parabolic" (22). The tip region is often an exception since there 𝑒𝑒 can be much smaller than 1; for example, 𝑒𝑒 = 0.12 at tip of the DW in Fig. 1(f). This region is called a "T-wall" (22) to stress that the polarizations 𝐏𝐏 1 and 𝐏𝐏 2 on opposite sides of the wall are tangential to it (and antiparallel to each other), Fig. 1(a). In the samples with colloidal seeds, the eccentricities deviate from 1 rather strongly, by ±0.25 , Fig. 1(d-g) and Fig. 2. An apparent reason is the meniscus around the colloidal spheres, which implies a nonzero dihedral angle between the surface of the sphere and the N F interfaces with air and glycerol in Fig. 1(d-g) and polystyrene-coated glass plates in Fig. 2. The resulting thickness gradients create a torque forcing 𝐏𝐏 to be perpendicular to the gradient direction, in order to avoid splay (43).
In solid ferroelectrics, the DW are often of the Ising type. For example, in a 𝜋𝜋-wall of the Ising type separating two domains with antiparallel polarizations, the polarization remains parallel to the same crystallographic axis, but its magnitude decreases to zero, |𝐏𝐏| → 0, in the middle of the wall (28). The parabolic DWs in our experiments are different from the Ising DWs as they do not show any significant decrease of the polarization magnitude, as revealed by strong birefringence observed within the entire width 𝑤𝑤~10 μm of the DW, Fig. 1(a). The optical retardance across the wall changes continuously and smoothly, Fig. 1(a), lacking the abrupt discontinuity expected in an Ising wall. Beside strong birefringence, another argument against a "polarization melting" within the wall is a high energy of such a melting, which can be estimated, following a similar approach proposed by de Gennes for the nematic-to-isotropic transition (11),
as 𝜕𝜕 𝐵𝐵 𝑇𝑇/𝑉𝑉 ≈ 6 × 10 6 J/m 3 , where 𝜕𝜕 𝐵𝐵 = 1.38 × 10 -23 J/K is the Boltzmann constant, 𝑇𝑇 ≈ 400 K is the approximate temperature of the phase transition from the N F to an antiferroelectric or paraelectric phase, and 𝑉𝑉 ≈ 1 nm 3 is the molecular volume. This energy density is much higher than the elastic energy density of reorientation of 𝐏𝐏 within the DW, estimated as
, where 𝐾𝐾 � is some average of the bend and splay elastic constants, taken for the purpose of this comparison to be of an exaggerated value 10 3 pN. Instead of complete polarization melting, the prevailing mode of connection of two neighboring polarization domains 𝐏𝐏 1 and 𝐏𝐏 2 is through realignment of the polarization, as in the Bloch and Néel walls of solid ferromagnets; some variation of the absolute value of 𝐏𝐏 should not be excluded.
The overall parabolic shape of the P-wall separating a domain with a uniform polarization 𝐏𝐏 1 and a domain with a circular polarization 𝐏𝐏 2 is explained by the avoidance of the bound charge on it, i.e., 𝜎𝜎 𝑏𝑏 = (𝐏𝐏 1 -𝐏𝐏 2 ) • 𝐤𝐤 ̂= 0, Fig. 3. The last condition is fulfilled when the angle 𝜃𝜃 1
between 𝐏𝐏 1 and the DW is equal to the angle 𝜃𝜃 2 between 𝐏𝐏 2 and the DW, 𝜃𝜃 1 = 𝜃𝜃 2 = 𝜃𝜃 , Fig. 3(a). These two angles increase as one approaches the vertex of the parabola, located at the origin (𝑥𝑥, 𝑦𝑦) = (0,0) of the Cartesian coordinates: 𝜃𝜃 1 = 𝜃𝜃 2 = 𝜃𝜃 = arctan�𝑥𝑥/𝑓𝑓 , where 𝑓𝑓 is the distance between the vertex and the focus, i.e., the center (𝑓𝑓, 0) of the circular domain, The inner structure of the parabolic DW is different far away from the vertex, where 𝑒𝑒 ≈ 1 and near it. Far away from the vertex, 𝐏𝐏 realigns from 𝐏𝐏 2 to 𝐏𝐏 1 by a small angle 𝛿𝛿 = 𝜋𝜋 -2𝜃𝜃, Figs.1(a), 3. As is easy to see, 𝛿𝛿(𝑥𝑥 → ∞) → 0 and the associated elastic energy is low. As ) ) Retardance (nm) 100 µm (e) μm ) y ) P A Silica Particles 50 µm (d) (f) (g) (d) (e) ) ) P A P A P A
(a) (b)
Solid ferroelectrics exhibit Néel DWs, in which the polarization experiences splay-bend realignment in the plane perpendicular to the wall, and Bloch DWs, in which the polarization twists along the axis perpendicular to the wall (28). The parabolic DWs in our study reveal splaybend deformations of 𝐏𝐏 in the plane of the sample, as discussed in the previous section, Figs. 1-3, but also twists.
The twists along the normal to the sample, i.e., along an axis in the DW plane, are easy to uncover under a polarizing optical microscope with decrossed polarizers, Fig. 5. In Fig. Alternating left and right-handed twists of 𝐏𝐏 around the 𝑧𝑧-axis normal to the film exist even in the geometry when a uniform unidirectional alignment, say, 𝐏𝐏 = (0, 𝑃𝑃, 0) corresponds to the Oseen-Frank elastic (but not the electrostatic) energy minimum. These twisted states are observed in films with one surface providing a unidirectional alignment and the opposite surface being azimuthally degenerate (as the surfaces in the present study) (44). Although the twists increase the elastic energy, they reduce the electrostatic energy (44).
Twists are also apparent in the T-walls, as these walls do not show complete extinction in the segments that are parallel to the polarizer or analyzer. This feature also suggests that the polarization does not form a homeotropic region 𝐏𝐏 = (0, 0, 𝑃𝑃) in the center of the wall, as would be the case for a pure Bloch wall. It is likely that the 𝑧𝑧-component of polarization varies with 𝑧𝑧 within the DW and thus introduces twist along a horizontal axis accompanied by splaybend near the interfaces, as described for 2𝜋𝜋 walls previously (19). The twist with a non-zero 𝑧𝑧component reduces the elastic energy of splay necessitated by the in-plane 𝜋𝜋 -realignments of 𝐏𝐏 (19). The small width of the T-walls makes it difficult to decipher the complex deformation field solely by optical microscopy. Nevertheless, it is safe to conclude that the T-and P-walls represent a complex mix of all three bulk deformations, splay, bend, and twist.
Twist deformations are a common feature of other DWs in the N F , as they have been previously identified in 2𝜋𝜋 (19) and 𝜋𝜋 (45) DWs. In the latter case, a 𝜋𝜋 DW that separates two antiparallel orientations of 𝐏𝐏 in a unidirectionally rubbed cell splits into two surface disclinations. These disclinations are shifted with respect to each other in the plane of the sample, which results in a twist around the normal to the sample. The splitting of DWs into surface disclinations can also be observed in conventional nematics in unidirectionally rubbed cells [40].
Whether the defect represents a wall or two surface disclinations depends on the balance of elasticity and in-plane anchoring, as explained by Kléman (46). To verify whether the DWs in our experiments can be split into pairs of surface disclinations, we created in-plane shears by shifting one plate with respect to the other. The shear does not result in separation of the DW images, neither of the P-walls, nor the T-walls, which suggests that these walls are not split into surface disclinations. The issue of walls vs. disclinations is briefly discussed below to clarify the reason for conflicting reports in the literature (19,45).
Consider a balance of surface anchoring and elasticity of an N F DW with a polarization realignment by 𝜋𝜋 from an in-plane easy direction, say, along the 𝑥𝑥-axis, to the direction (-𝑥𝑥). In the N F , in-plane anchoring is polar (12) and can be described by a potential 𝑊𝑊(𝜑𝜑) = 2.4. Walls in freely suspended films. The texture of freely suspended films of DIO show that the polarization 𝐏𝐏 is parallel to the edges of the square opening, forming bend DWs along the diagonals, Fig.6. This arrangement is supported by the geometrical anchoring effect of the meniscus and by a strong tendency of 𝐏𝐏 to align tangentially to any N F interface, which avoids a strong surface charge. Even a small tilt 𝜓𝜓~5 o of 𝐏𝐏 from the 𝑥𝑥𝑦𝑦 plane of a bounding plate or an interface would produce a surface charge density 𝑃𝑃 𝑧𝑧 ~𝑃𝑃𝜓𝜓 ~6 × 10 -3 C m -2 , which is larger than the typical surface charge (10 -4 -10 -5 ) C m -2 of adsorbed ions reported for the N (47, 48). Recent experiments (49) with curved capillaries filled with the N F and subject to a longitudinal electric field provide firm evidence of a strong tangential anchoring at the N F interfaces. As in the case of the parabolic DWs, the diagonal DWs are of a finite width, stabilized by the electrostatic effect. As described by the models below, disparity of the elastic constants is capable of reproducing the finite DWs width. (a) (b) DIO;
In this section we will propose a mathematical model which sits within a standard approach to describe equilibrium configurations of nematic liquid crystals. A key aspect of our model is an assumption that the energetic cost of splay deformations far exceeds those of bend and twist. With such a theory in hand, we can capture a wide variety of experimentally observed morphologies. Success in this endeavor should also enable a reverse process of predicting material parameters of ferroelectric nematics on the basis of experimental observations.
As was already alluded to in the introduction, in modeling the experimental set-up, we propose an energy based on a vector-valued order parameter 𝐏𝐏 representing the local polarization of the ferroelectric nematic sample. We then pursue analysis of an Oseen-Frank type of energy. However, in light of the fact that polarization may vanish in the neighborhood of defects or walls, we invoke a Landau-de Gennes (or Ginzburg-Landau) type of potential favoring unit vectors as opposed to a hard constraint that |𝐏𝐏| = 𝑃𝑃 0 everywhere in the sample. Here 𝑃𝑃 0 denotes a preferred value for the magnitude of the polarization vector.
It would also be natural to model the experiments discussed in this article using an energy based on both polarization and a nematic director, but as we shall indicate through numerous computational experiments in the subsequent section, our model based solely on 𝐏𝐏 = (𝑃𝑃 (1) , 𝑃𝑃 (2) , 𝑃𝑃 (3) ) already successfully captures an array of morphologies that emerge in the laboratory.
Given that the experiments are carried out on a thin domain, we take as our sample the set
where 𝐿𝐿 = 𝑂𝑂(1) and 0 < ℎ ≪ 1. Then for 𝐏𝐏: Ω → ℝ 3 we introduce the (dimensional) elastic energy
This is the analog of the Oseen-Frank energy for nematic liquid crystals written for polarization where we ignore electrostatic interactions. It is supplemented with the penultimate potential term fixing the preferred value for the magnitude of the polarization vector.
To explain the final surface anchoring term appearing in Eq. ( 2), we note that at the interface of an apolar nematic and an isotropic fluid, the only angular dependence that could enter the surface energy density is via a term proportional to (𝐧𝐧 � ⋅ 𝛎𝛎 �) 2 , where 𝐧𝐧 � is the director and 𝛎𝛎 � is the normal to the surface. For the anisotropic surface tension 𝜎𝜎 𝑁𝑁𝑁𝑁 to yield a minimum at some "easy cone" 0 ⩽ 𝜑𝜑 𝑒𝑒𝑒𝑒 ⩽ 𝜋𝜋/2 , the term (𝐧𝐧 � ⋅ 𝛎𝛎 �) 2 should be supplemented by higherorder terms,e.g.
(𝐧𝐧 � ⋅ 𝛎𝛎 �) 4 + 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐, which can be rewritten as
as long as -1 ≤ 𝛾𝛾̅ /𝛾𝛾 ≤ 0. Here 𝐧𝐧 � 𝑒𝑒𝑒𝑒 is the "easy axis" making the equilibrium angle 𝜑𝜑 𝑒𝑒𝑒𝑒 with 𝛎𝛎 �, defined from cos 2 𝜑𝜑 𝑒𝑒𝑒𝑒 = -γ �
. If 𝐧𝐧 � 𝑒𝑒𝑒𝑒 is tangential to the interface, then 𝜑𝜑 𝑒𝑒𝑒𝑒 = 𝜋𝜋/2 and 𝐧𝐧 � 𝑒𝑒𝑒𝑒 ⋅ 𝛎𝛎 � = 0, thus
A similar consideration is valid for the ferroelectric nematic with 𝐏𝐏 replacing 𝐧𝐧 � and 𝛎𝛎 � = (0,0,1) as in Fig. 7. This justifies the term with By setting
we arrive at the following simplified expression for the elastic energy
This version of the model captures the experimentally observed features of ferroelectric nematics. Since a key feature of the ferroelectric nematics under consideration here is the relatively high energetic cost of splay, we assume that 𝑀𝑀 ≫ 𝐾𝐾 where we note that in the second term above we are effectively folding the cost of twist and bend into one equal constant term.
Because of the degenerate planar anchoring, the angle between the polarization and any bounding surface is essentially zero and, therefore, the anchoring strength 𝛾𝛾 ≫ 1.
Taking a characteristic diameter of Ω to be, say 𝐿𝐿, we identify two small dimensionless parameters:
with two more dimensionless parameters
)
for a polarization vector 𝐏𝐏 defined for (𝑥𝑥, 𝑦𝑦) ∈ Ω 0 and 𝑧𝑧 ∈ (-
). Here Ω 0 is a square of unit side length and the subscripts 𝑥𝑥, 𝑦𝑦, and 𝑧𝑧 denote derivatives with respect to these variables.
We seek minimizers of 𝐸𝐸 with bounded energy. Because 𝛿𝛿 and 𝜀𝜀 are small, while 𝜇𝜇 and Γ are both 𝑂𝑂(1), we observe that dependence of 𝐏𝐏 on the variable 𝑧𝑧, the nonvanishing third component of 𝐏𝐏 and deviations of |𝐏𝐏| from 1 all incur very high energy cost. In light of the first fact, in what follows we make a further simplification that the polarization 𝐏𝐏 is independent of 𝑧𝑧, leading to the reduced energy
Now, the behavior of a configuration that minimizes Eq. ( 5) is dictated by relative sizes of 𝜀𝜀 and 𝛿𝛿.
Suppose first that 𝛿𝛿 ≪ 𝜀𝜀 . In this case we can conclude that the last term in Eq. ( 5)
dominates unless the third component of 𝐏𝐏 vanishes. Therefore, we can impose the condition that 𝐏𝐏 lies in 𝑥𝑥𝑦𝑦-plane. Under these assumptions, to leading order we find that the resulting energy ℰ 𝜀𝜀 can be described through a two-component vector field p= (p (1) (𝑥𝑥, 𝑦𝑦), p (2) (𝑥𝑥, 𝑦𝑦))
via
The assumption of 𝑧𝑧 -independence aligns with the experimental observations carried out in thin samples sandwiched between two interfaces. However, interestingly enough, numerical simulations and formal asymptotics indicate that this reduction from Eq. (6) to Eq. ( 7)
is far from being mathematically straightforward and thus will not be discussed here. Now, instead let 𝜀𝜀 ≪ 𝛿𝛿 . Then the integral of
(|𝐏𝐏| 2 -1) 2 will be the largest contributor to the energy 𝐸𝐸 𝑟𝑟 so that we are justified to assume that �𝐏𝐏 � � ≡ 1 in Ω 0 . Writing 𝑃𝑃 (3) in terms of other components of 𝐏𝐏 then gives
We also have 𝑃𝑃 𝑥𝑥 (1) + 𝑃𝑃 𝑦𝑦 (2) = d𝑖𝑖𝑖𝑖 𝐩𝐩, because 𝐩𝐩 is independent of 𝑧𝑧. Substituting these expressions into (6), we obtain the energy
where
We observe that the expressions for Eq. ( 7) and Eq. ( 8) are mathematically very similar even though the potential terms originate from two unrelated sources. The differences arise from the presence of the gradient of the third component in Eq. ( 8) and different relationships between the parameters. From the physical perspective, the energy in Eq. ( 8) allows for twist deformation-in particular, within a wall-while Eq. ( 7) only permits splay and bend. In the next subsection we discuss asymptotics for Eq. (7). Then, in the following section devoted to numerics, we will present examples of energy-minimizing configurations for both Eq. ( 7) and Eq. (8).
In this section we indicate how to mathematically analyze energy minimizing configurations for Eq. ( 7), taking advantage of the smallness of 𝜀𝜀 . In particular, we describe analytical techniques for constructing elastic walls for given anchoring conditions on the twodimensional polarization 𝐩𝐩. This subsection aims to provide the mathematical justification for the numerical results presented in the next section.
The energy in Eq. ( 7) penalizing splay over bend has been analyzed in (50) . The most salient observation in (50) is that, as 𝜀𝜀 → 0, the minimization problem (7) approaches a sum of bulk splay cost and a wall cost given by
where 𝐩𝐩 has prescribed values on 𝜕𝜕Ω 0 and satisfies |𝐩𝐩| = 1 everywhere in Ω 0 . The symbol 𝐽𝐽 𝐩𝐩 represents the domain wall, that is, a curve across which 𝐩𝐩 jumps in order to save on the cost of splay, and 𝐩𝐩 + and 𝐩𝐩 -denote the values of the polarization on either side of the wall.
A mathematical subtlety we wish to highlight with regard to the energy ℰ 0 is that eligible vector fields 𝐩𝐩 that exhibit jump discontinuities across such a curve 𝐽𝐽 𝐩𝐩 must nonetheless respect the integration by parts formula (i.e. the Divergence Theorem). This induces a requirement that the normal component of 𝐩𝐩 remains continuous across the wall. Since the potential term in Eq. ( 7) forces the polarization field 𝐩𝐩 to be of a unit magnitude away from the wall for 𝜀𝜀 ≪ 1, it follows that the tangential component of 𝐩𝐩 simply switches sign on either side of the wall. Thus, we see that the physical requirement that a domain wall be uncharged, as discussed in the introduction, is manifested in our model through an application of integration by parts.
A simple example that illustrates the utility of this continuity condition is in order.
Suppose that a wall separates two distinct states with zero splay, for example, a state where 𝐩𝐩 ≡ 𝑐𝑐𝑐𝑐𝑛𝑛𝑐𝑐𝑐𝑐 and a state where 𝐩𝐩 has a circular vortical pattern. In view of Eq. ( 9), then all of the energy of such a configuration will be concentrated on the wall. Then the placement of the wall is dictated by the requirement that across it, the tangential component of polarization switches sign.
To be specific, suppose that the constant state is 𝐩𝐩 = (1,0) and the divergence-free
Let us describe the wall in terms of its distance, say 𝜌𝜌(𝜃𝜃), from the origin which serves as the center of the vortex. That is, as a function of the polar angle 𝜃𝜃, suppose the wall is given by 𝜃𝜃 ↦ 𝜌𝜌(𝜃𝜃)(cos𝜃𝜃, sin𝜃𝜃).
Then a tangent vector to the wall is given by
and the condition of sign-switching tangential component and continuous normal component can be expressed as
This equation simplifies to the separable ODE
which can be readily solved to find that the wall is given by
Observing that Eq. ( 11) is the polar equation of a parabola, we see that a these two divergencefree states must be separated by a parabolic domain wall, also known as a P -wall, as already discussed in the introduction and in the experimental part. Numerical simulations conducted through minimization of ℰ 𝜀𝜀 for 𝜀𝜀 small confirm that this geometry emerges for appropriate boundary conditions. What is more, this arrangement conforms with the expeccted emergence of conics dicussed in the introduction and is consistent with experimental observations as well, giving support for the validity of the model (cf. (20), Fig. 7).
Another configuration consistent with splay-free bulk, and one that emerges experimentally as well, is the appearance of domain walls in a triple junction configuration. In the simplest scenario, consider three rays, meeting at the origin and separated from each other by an angle of 120 ∘ , i.e., at angles 0 ∘ , 120 ∘ and 240 ∘ with the 𝑥𝑥 -axis. Within each 120 ∘ sector, place a uniform state so that the director makes an angle of 150 ∘ with the 𝑥𝑥-axis in the first sector, an angle of -90 ∘ with the 𝑥𝑥-axis in the second sector and an angle of 30 ∘ with the 𝑥𝑥-axis in the third sector. Just as in the previous example, the entire energy of this state is concentrated on the walls and the configuration respects the continuity condition of the normal component of polarization across all walls. Another example comes from exchanging the constant states from the previous example with three circular vortices. Once again, all of these configurations may be observed experimentally.
One can also use Eq. ( 9) to construct walls in more complicated settings where, for example, the divergence does not vanish on either side of the walls; again see (50). For this pursuit, we observe that a minimizer of ℰ 0 satisfies the criticality condition 𝐩𝐩 ⊥ ⋅ ∇ (div 𝐩𝐩) = 0 𝑖𝑖𝑛𝑛 Ω 0 \ 𝐽𝐽 𝐩𝐩 , where 𝐩𝐩 ⊥ = (-p (2) , p (1) ),
along with the requirement |𝐩𝐩| = 1 . Writing 𝐩𝐩 locally as 𝐩𝐩(𝑥𝑥, 𝑦𝑦) = (cos𝜃𝜃(𝑥𝑥, 𝑦𝑦), sin𝜃𝜃(𝑥𝑥, 𝑦𝑦))
and defining the scalar
one has that Eqs. (11)-( 12) is equivalent to the following system of partial differential equations for the two scalars 𝜃𝜃 and 𝜂𝜂:
-sin𝜃𝜃 𝜃𝜃 𝑥𝑥 + cos𝜃𝜃 𝜃𝜃 𝑦𝑦 = 𝜂𝜂,
-sin𝜃𝜃 𝜂𝜂 𝑥𝑥 + cos𝜃𝜃 𝜂𝜂 𝑦𝑦 = 0.
This system of first order partial differential equations has a fairly simple solution obtainable by the method of characteristics, cf. (51). Starting from any initial curve in Ω 0 parametrized via
This configuration was obtained numerically in (50) using the finite elements package COMSOL (52) by solving the full system of partial differential equations that describe the critical points of Eq. (7). In this case, the walls have a simple shape of a cross, but the morphology of the polarization field is fairly complex. Nonetheless, the same solution can be constructed analytically by using the method of characteristics described above. The corresponding plots are shown in Fig. 10 for the top right quarter of the rectangle and they clearly have the same behavior as the numerically derived result in Fig. 9. is a solution obtained using characteristics.
In this section, we demonstrate that the minimizers of the two-dimensional energy in Eq. To benchmark our model, we will begin by looking for an optimal configuration of Eq. (7) in a rectangular domain with boundary conditions corresponding to a circular polarization on the It is well known that the energy minimizer in this case is characterized by the presence of a parabolic wall, Fig. 11 that indeed forms in the domain during the gradient descent for Eq. (7).
This outcome also corresponds to the analysis in the previous section, cf. Eq. (11).
Next, we will consider inclusions that have sizes comparable to the thickness of the film.
To this end, consider the situation shown in Fig. 12. Here, a spherical inclusion penetrates the top and the bottom of the film at the interfaces with air and glycerol and therefore a meniscus forms on each boundary. As alluded to previously, because the polarization vector 𝐏𝐏 wants to remain parallel to these interfaces, in order to minimize the elastic energy in three dimensions 𝐏𝐏 orients along the normal to the thickness gradient. and 𝛿𝛿 = 0.12.
Therefore, we expect that the polarization should be pinned in a circular pattern around the inclusion on both surfaces of the film, cf. Fig. 12. In order to model a polarization field in a ferroelectric nematic film with inclusions within the 2𝐷𝐷-framework, we can then excise a disk corresponding to the inclusion along with the meniscus and impose tangential anchoring on the boundary of the disk.
Now we consider a rectangular domain with a hole representing an excised disk around an inclusion. We impose constant boundary conditions 𝐏𝐏 = (1,0,0) on the boundary of the rectangle and tangential boundary conditons, 𝐏𝐏 = (-sin𝜃𝜃, cos𝜃𝜃, 0) on the boundary of the disk for a polar angle 𝜃𝜃 with respect to the center of the disk. We then find the energy-minimizing configuration of polarization by using steepest descent for the energy Eq. ( 8), as shown in Fig. 13.
We observe a parabolic downward-pointing wall with a faint upward parabolic "ghost" wall.
These features have also been observed in experiments, Fig. 4. Note that in this case the elastic deformation at the tip of the wall is dominated by twist around the axis in the 𝑥𝑥𝑦𝑦-plane. As already discussed, such a twist under the condition of 𝑧𝑧 -independent polarization might be prevented by electrostatics at most interfaces. Experiments suggest that the details of the fine structure of domain walls involve all three types of deformations, including variations along the 𝑧𝑧-axis, which reconcile the twist in the bulk with tangential anchoring at the boundaries (19).
However, we do not exclude a possibility that in the future a homeotropic or strongly titled anchoring might be achieved at some N F -substrate interfaces, in which case such a twist across the entire thickness of an N F slab could be observed.
Figures 14 and 15 confirm that our modeling approach also produces the correct behavior in the system consisting of two particles imbedded in a ferroelectric nematic film. Figures 1(d-g)
show the corresponding experimental image. The plot in the Figs. 14 and 15 are, respectively, minimizers of the energy ( 8) and ( 7) in the region exterior to these disks. It is apparent that, while the wall morphologies in the two figures match experimental observations, there are some subtle differences, e.g., the secondary walls are present only in Fig. 15. indicates the value of 𝑃𝑃 (3) , while the arrows represent the projection 𝐩𝐩 of polarization onto the plane of the film. Fig. 15. Morphology of a ferroelectric nematic film with two inclusions: an energy minimizing configuration for the energy Eq. (7). Here 𝜇𝜇 = 8 and 𝜀𝜀 = 0.004. The color indicates the value of |𝐏𝐏|, while the arrows represent the polarization 𝐏𝐏 = �𝑃𝑃 (1) , 𝑃𝑃 (2) , 0�. Note that here 𝐏𝐏 = 𝐩𝐩.
The model Eq. ( 7) is based on a two-component polarization vector and therefore the walls in Fig. 15 are of the bend-splay type. On the other hand, because the model Eq. ( 8) involves three components, the corresponding wall structure is allowed to exhibit twist and indeed, this is what occurs in Fig. 14 as can be discerned from the deviations of the third component of 𝐏𝐏 away from zero in the interior of the wall. In Fig. 16 we show a similar configuration for four inclusions. and 𝛿𝛿 = 0.12.
The plot on the right in Fig. 17 is obtained by using steepest descent of the energy Eq. ( 8)
starting from initial data with regions of both clockwise and counterclockwise twist of the polarization. This plot shows a single inclusion with an associated parabolic wall. However there is a difference with Fig. 13 in that the left half of the wall has positive twist, while its right half has negative twist. Suppose that the polarization field satisfies periodic boundary conditions on the vertical sides of the strip. Starting from initial data containing twist of both signs and using gradient descent leads to a local minimum of the energy in Eq. ( 8) with a wall along the axis 𝑦𝑦 = 0 such that the twist alternates as one moves along the wall, Fig. 18. Once again, the intervals of opposite twist are separated by point defects.
Finally, to model freely suspended films of DIO, we consider the energy (7) over a square domain where the direction of the polarization vector on the boundary is parallel to the boundary itself. The results are shown in Fig. 19 and demonstrate formation of bend-splay domain walls along the diagonals of the square, which one can compare favorably to the experimental findings in Fig. 6.
As compared to their solid counterparts, ferroelectric nematics show a rich variety of polarization Except near the tip of the DW in azimuthally-degenerate slabs, the eccentricity is close to 1, hence the name "P-wall". The P-wall structure is complex, with the bend and splay of 𝐏𝐏 in the (𝑥𝑥, 𝑦𝑦) plane of the film and a twist of 𝐏𝐏 along the normal 𝑧𝑧 to the film. Along the P-wall, the left-and right-handed twists alternate, being separated by defect lines along the 𝑧𝑧-direction.
As one approaches the tip of the DW, the realignment trajectory of 𝐏𝐏 starts to resemble a sharp U-turn with a high splay-bend energy; the geometry of the P-wall is replaced by the new structure, called the T-wall, in which the polarization vectors of the two domains are both tangential to it. The T-branch is separated from the P-branches by two -1/2 disclinations. The Tbranch is more narrow than the P-branches, which might be explained by a smaller amount of splay in it; the prevailing deformation appears to be the twist along the horizontal and vertical axes with some elements of splay-bend near the bounding plates. Since the T-wall embraces the circular domain on the inside, its eccentricity is less than 1; the curved T-wall causes bending of 𝐏𝐏 in the outside domain, which produces a "ghost" parabola.
We show that these experimentally observed shapes are well described by a version of the Oseen-Frank energy functional in which the splay elastic constant is much larger than the twist and bend constants. Whenever the third component of the polarization vector is set to vary, the model allows for the presence of both twist-and bend-splay DWs. This approach recovers principal features of ferroelectric nematic morphologies, such as the secondary "ghost" domain walls and mixed handedness of the twist of the polarization vector along the DW. Within the modeling, the finite width of domain walls is attributable to the large disparity between the elastic constants, in that the system accommodates the splay by forming structures with strong bend and/or twist. The finite width of domain walls in the model is fully supported by the experiments, as evident from the comparison of parabolic walls produced by simulations in Figs. We also note that model (7) allows for bend and splay within the walls, while (8) incorporates all elastic modes, including twists, yet both produce the same DW morphology and director distribution outside of the DWs. At the same time, within the walls, the director configurations are different and are dominated either by bend or by twist. Further, note that in numerical experiments ghost walls seem to appear only in the bend dominated walls, therefore conceivably one could try to use the presence of ghost walls in physical experiments as an indicator of a bend deformation within DWs. Needless to say, the models described in this work
are not uniformly applicable in that they fail to capture all the features of ferroelectric nematic configurations when the dependence on the 𝑧𝑧-coordinate cannot be ignored. In particular, this shortcoming applies to the details of the wall structure that in fact might be three-dimensional.
The present model also does not capture the expected presence of double layers of bound charges of a density -𝜕𝜕𝑃𝑃 𝑘𝑘 /𝜕𝜕𝜕𝜕 embracing the parabolic DWs, although these are visible around straight DWs in Fig. 9a. The full three-dimensional inner structures of DWs will be addressed in a future investigation.