Evaluation of polar alignment structure and surface anchoring energy in ferroelectric nematic liquid crystals by renormalized transmission spectroscopic ellipsometry

It is essential to estimate the surface polar anchoring energy in order to discuss the interfacial orientation of ferroelectric nematic liquid crystals. In this study, we accurately estimated the twist angle of a π -twist cell with an antiparallel rubbing manner, by means of renormalized transmission spectroscopic ellipsometry, which has a great deal of experience in measuring the twist angle of ordinary nematic liquid crystals. We also succeeded in estimating the reduced polar anchoring energy from the essential equation derived from the simplified torque balance equation. It is assumed that the reduced surface polar anchoring energy is on the order of 102.

It is essential to estimate the surface polar anchoring energy in order to discuss the interfacial orientation of ferroelectric nematic liquid crystals.In this study, we accurately estimated the twist angle of a p-twist cell with an antiparallel rubbing manner, by means of renormalized transmission spectroscopic ellipsometry, which has a great deal of experience in measuring the twist angle of ordinary nematic liquid crystals.We also succeeded in estimating the reduced polar anchoring energy from the essential equation derived from the simplified torque balance equation.It is assumed that the reduced surface polar anchoring energy is on the order of 10 2 .© 2024 The Author(s).[4][5] In addition to its high dielectric constant and agile response under the external electric field, FNLCs are particularly noteworthy because, in common sense, ferroelectricity and fluidity were thought to be incompatible properties.[13][14][15] FNLCs exhibit a polar molecular alignment different from usual nematic liquid crystals (NLCs), with regards to the director n, it is known that n and −n are not equivalent from the viewpoint of the symmetry.The origin of the polar molecular alignment is interpreted to be due to the low structural symmetry of the liquid crystal molecules. 5)dditionally, it has been reported that FNLCs form a variety of polar alignment structures in bulk due to the spontaneous polarization of the molecules and their interactions with the substrate surface. 6,16)Consequently, the high polarity of FNLC molecules competes with the polarity of the alignment surfaces, leading to the emergence of various twisted and planar alignments.[19][20] However, a quantitative method for evaluating such surface anchoring energy for FNLCs has not yet been established.
In this study, a self-consistent explanation for the relationship between the polar alignment structure and surface alignment of FNLCs will be proposed, where the theoretical framework for quantitative evaluation of the surface polar anchoring energy is well-grounded. 21,22)Here, it may be helpful for the reader to research a supplementary explanation of the technical term "surface polar anchoring," which is sometimes recognized as a term referring to out-of-plane directional anchoring.In this paper, we will discuss the directional and/or electrical polarity in which the easy-toalign axis is tilted within the plane of the substrate, which is called the "polar angle," and it is assumed that molecules do not tilt in the out-of-plane direction.This is customary in papers dealing with FNLCs. 18,19)The torque balance equation indicates that a quantitative evaluation of the surface polar anchoring energy is possible by experimentally determining the angle j i between the easy-to-align axis and the rubbing direction.][25] The goal is to demonstrate that the surface polar anchoring energy of FNLCs can be experimentally determined by applying the RTSE that has already been established for NLCs.
Suppose that the FNLCs were twisted by 180°(i.e.p-twist) inside a sandwich-type cell with antiparallel rubbed inner surfaces.Figure 1 shows the conceptual Cartesian laboratory coordinate inside the sample, where the twist axis is parallel to the z-axis which is normal to the substrate.Here, the rubbing direction is set parallel or antiparallel to the x-axis.When the surface anchoring does not satisfy the ideal strong anchoring condition in the actual case, the easy-to-align axis of the FNLC molecules in the vicinity of the surface does not coincide with the rubbing direction due to the elastic torque in the bulk.Here, the deviation angle of the easy-to-align axis from the rubbing direction is expressed as j .
i Such logic is typical for the consideration based on a continuum theory for the conventional nematic phase. 21)The torque balance equation considering polar anchoring is detailed in our previous paper. 26)The polar direction of each FNLC molecule in the vicinity of the top and bottom substrate tends to align parallel to the rubbing direction so as to couple with the surface polarity.[19] By ignoring the apolar surface anchoring energy, the relationship between j i and the surface polar anchoring energy W p can be expressed as follows, where the deviation angles at the upper and the lower surfaces are both assumed to be j i due to the symmetry, and the rubbing alignment directions at the lower and upper surface are set to 0 and π, respectively.Assuming a simple elastic twist deformation, the elastic energy F B can be determined utilizing an adaptation of a usual manner of the elastic theory for nematic LCs, which is expressed by the following form, where K T denotes the simplified elastic constants for twist deformation, and d is the FNLC layer thickness.Here, the right-handed twist is taken as positive.Consequently, the total free energy per unit area of the FNLC cell can be formulated as the summation of F B and F S for the upper and the lower surfaces, By minimizing the total free energy F with respect to j , i following equations are derived; In Eq. ( 5), / denotes the reduced polar anchoring energy at the surface.This standardized and dimensionless quantity is derived by dividing the surface anchoring energy with the elastic constant and the reciprocal of the FNLC layer thickness, which provides a substantive parameter elucidating the underlying physics.An advantage of discussing based on the reduced surface polar anchoring energy W n is that the relationship between W n and j i can be discussed without measuring K T which cannot be measured easily like a twist elastic constant K 2 in an ordinary nematic phase.This essential equation uniquely determines the relationship between the dimensionless reduced surface polar anchoring energy W n and deviation angle j .
i Figure 2 represents the characteristic curves of W n versus j .
i As is simply seen in Fig. 2, it is beneficial that W n can be uniquely determined if only j i can be measured experimentally.
We have conceived the RTSE that we have fostered as a method for determining the azimuthal anchoring energy of ordinary twisted nematic cells would be a suitable method for measuring j .i 24) The RTSE measurements were performed using a standard spectroscopic ellipsometer (M-150, Jasco Co., Japan) commercialized based on the photo-elastic modulator method, with the positional relationship between the sample stage and each optical component in the coordinate system as shown in Fig. 3.The FNLC sample is placed on the sample holder so that the transmitting ray is set perpendicular to the sample surface and the polarizer is adjusted to be almost parallel to the rubbing direction on the substrate surface.The ellipsometric angle Y (amplitude ratio) and D (phase difference) are generally defined by, 27)  respectively, and ] can be determined 031001-2 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd using the polytope method, 28) one of the common multivariate analysis methods.Finally, j i is obtained by subtracting j t from the angle formed by the antiparallel rubbing direction.
The specific procedure to determine j i employing RTSE is as follows.Before filling the FNLC substance, the gap of the sandwich cell d empty is determined by the optical interference method, and the nominal gap was 5.2 μm in this experiment.d empty is substituted as an initial value of the FNLC layer thickness in an optical simulation using the Jones matrix method.One of the issues is that antiparallel rubbing results in right-handed and left-handed twists with equal probability.In this experiment, several cells were fabricated in which only a right-handed (or left-handed) twist occurred by cell assembly in which the rubbing direction deviated several degrees from 180°, as shown in Fig. 1, where the two glass substrate assembling angles between the upper and lower substrates is indicated by a [deg.]The FNLC material used in this study is FNLC-919 (Merck), which was reported to exhibit the FNLC phase at RT (20 °C) 17) and has been utilized in fundamental and applied research fields. 29)The initial value of the birefringence of FNLC-919 was set as are determined for adjacent wavelengths by the procedure described above, where the ( ) l Dn can be well approximated by Cauchy dispersion and its coefficients were also determined at the same time.Searching for the multivariate that minimizes the difference between measured and simulated values with respect to ( ) l D and ( ) l Y , based on the polytope method as the best method for multivariate analysis, finally j t can be determined.A decrease in j t concerning the two-glass substrate assembling angles is equivalent to doubling j .i The procedure to fabricate the sandwich-type FNLC cell is as follows.First, soda-lime glass substrates whose surfaces were pre-coated with ITO thin films as electrodes were prepared.After the cleaning process, a polyimide (PI) solution for planar alignment whose nominal pretilt angle was less than two degrees was spin-coated on the surface of the ITO film, and then baked at 180 °C for 1 h to remove the solvent.Equipment with a roller covered by cotton cloth was utilized to rub the surface of the PI film unidirectionally.To promote planar alignment of the FNLCs, the rubbing directions of the two substrates were set to be almost antiparallel, ensuring that the rubbed PI surfaces face inward.As mentioned, a ranged from 174°to 180°, and the two-glass assembling manner was reversed in some sample cells (i.e. a = −175°, 176°and −180°).The gap between the two glass substrates was maintained using resin beads consisting of UV-curable adhesive (Photorec A-784-400, Sekisui Chemical Co., Ltd).The FNLC substance was filled into the empty cell through the capillary action at its isotropic phase temperature (>80 °C).During the cooling process after filling the substance, minimizing the temperature difference between the upper and lower substrates contributed to preserving the director alignment symmetry throughout the cell. 30,31)igure 4 represents the experimentally measured ( ) l D and ( ) l Y at 18 °C, where the ferroelectric nematic phase was confirmed by POM and a was 174°.In this experiment, ( ) l D and ( ) l Y measurements with the upper substrate as the incident side [Fig.4(a)] were followed by those measurements with the lower substrate as the incident side [Fig.4(b)] to reconfirm the symmetry of the twist angle.It is quite natural that, as shown in Fig. 4, there is no significant difference between the two measurements.The two solid lines in Fig. 4 are fitting curves from the numerical  simulations.The FNLC layer thickness, twist angle j , t and deviation angle j i obtained from the numerical fitting procedures are listed in Table I.The fact that the obtained d and j t are not far from the original design indicates the usefulness of this analysis method.The Cauchy dispersion coefficients obtained in the analysis process are also shown in Table I, and obtained ( ) l Dn is shown in Fig. 5, where ( ) / / It is found that the obtained ( ) l Dn is in good agreement with that reported by Yu et al. 17) It is beneficial to confirm the reproducibility of ( ) l Dn for the different methods carried out by another group.The significant cell thickness dependence of j i was not found while a was varied.The fact that j t and j i also have opposite signs when α was a negative sign experimentally shows the consistency of this model.It has been pointed out that there is a relationship between the rubbing direction and polarization, 17,22) and it is natural that the bulk twist takes the sense that lowers the total free energy.Lastly, j i is substituted into Eq.( 5).There is no doubt that surface polar anchoring is certainly present since the twist alignment occurs in the bulk.From the characteristic curve for the dependence of j i on W n as shown in Fig. 2, it is assumed that the reduced surface polar anchoring energy is on the order of 10 2 .Assuming K T = 10 −11 N, 19,32) the polar anchoring is estimated to be in the order of 10 −4 J/m 2 .We conclude that is meaningful to know the magnitude of W n from this study.Since the materials and measurement methods used are different, there is no significance in making direct comparisons with those reported by other groups, 6,19) however the results obtained in our study may serve as a starting point for future discussions.
In summary, fundamental methodology and experimental evaluation method that were focused on the surface anchoring effects were proposed.The characteristic curve of the relationship between the reduced surface polar anchoring energy and the deviation angle from the rubbing direction for FNLCs filled in the sandwich cell with antiparallel rubbing, which was derived from the classical torque balance equation with surface anchoring energy function, was presented.The optical measurement by means of the ellipsometry for the actual twist angle of FNLC cells showed that the deviation angle can be evaluated experimentally.Significantly, it was experimentally confirmed that the reduced surface polar anchoring energy is on the order of 10 2 .Until now, the insufficiency of evaluation methods for surface anchoring energy has in part held up specific FNLC device designs.This evaluation method is expected to facilitate future progress in device application utilizing FNLCs.031001-4 © 2024 The Author(s).Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd and r s denote the amplitude of the total emergent electromagnetic plane waves by the p-polarized (or s-polarized) incident light moving toward the subject.D and Y for the incident light with wavelength l [i.e. ( ) l D and ( ) l Y ] are quite sensitive to d and the twist angle j .t By fitting the experimentally measured ( ) l D and ( ) l Y with the optical simulation based on the Jones matrix method with appropriately supposed birefringence ( ) l Dn , d, and j t as well as the refractive indices for ordinary and extra-ordinary rays [

Fig. 1 .
Fig. 1.The model of the cell used in the experiment.
referring to the value measured by Yu et al. using a Berek compensator.17)The FNLC layer thickness d was determined through numerical fitting and was obtained to be 5.2 μm, which coincided with d .empty In the beginning of the numerical simulation, a tentative value of D at l= 589 nm was calculated using d .empty Next, the tentative value of Ψ was determined by combining the Jones matrix method.It should be noted that the effects of multiple reflections and multiple beam interferences caused by indium tin oxide (ITO) and alignment layers as optical components are not taken into consideration in this experiment.Continuously, ( ) l D and ( ) l Y

Fig. 2 .
Fig. 2. Correlation between deviation angle and the reduced surface anchoring energy.

Fig. 4 .
Fig. 4. Measurement data and fitting results for phase difference Δ and amplitude ratio angle Ψ using the Jones matrix calculus.(a) Measurement and analysis results when light is incident from the front.(b) Measurement and analysis results when light is incident from the back.

Table I .
Results of cell parameters for both surfaces and verification of data consistency.Experimentally obtained Cauchy's coefficients of FNLC-919 are also shown for each substrate assembling angle a.