# Spiral Pitch Control in Cholesteric Liquid Crystal Layers with Hybrid Boundary Conditions

^{*}

## Abstract

**:**

## 1. Introduction

_{||}, pn

_{⊥}), where p is the spiral pitch, and n

_{||,⊥}are the main refractive indices of the CLC. Obviously, the control of the spiral pitch is a key task when it comes to changing the spectral position of the stop-band in order to use the CLCs in photonic applications. Of particular interest are the CLC systems in which the pitch changes controllably under different kinds of external influences [13,14,15,16,17]. For example, it has been demonstrated in [16,17] that the spiral pitch can be controlled in wedge-shaped CLC cells for tuning the lasing wavelengths. However, in case of a fixed concentration of the chiral dopant, the tuning range is rather short (of about 10 nm on the wavelength scale), and the design of the wedge-structured CLC film with the spatial gradient of the chiral dopant is proposed for expanding the tuning range [17].

## 2. Materials and Methods

_{0}≅ 0.55 μm.

_{p-p}) in the packets was varied in increments of 50 V up to 300 V. All the spectra were measured in unpolarized light for beam propagating along the z-axis (Figure 1a), so the surface with the electrodes was located closer to the light source.

## 3. Results and Discussion

#### 3.1. Experimental Optical Transmittance Spectra

#### 3.2. Analytical Model

_{1-3}are the moduli of elasticity; d is the thickness of the CLC layer; q

_{0}and q are wave numbers that determine, respectively, the natural pitch of the spiral p

_{0}= 2π/q

_{0}and the pitch realized in the CLC layer p = 2π/q; φ

_{s}

_{1}, θ

_{s}

_{1}and φ

_{s}

_{2}, θ

_{s}

_{2}are, respectively, the azimuthal and zenithal angles of the director at the first and second boundaries of the layer; W

_{1}and W

_{2}are anchoring potentials, respectively, at the first and second boundaries.

_{s1}) and second (φ¬

_{s2}) boundaries and the wavenumber q. This relationship can be represented as

_{s1}= 0, θ

_{s1}= π/2) with an infinite anchoring energy (rigid boundary conditions), and at the second boundary there is a homeotropic alignment with a relatively weak anchoring energy W

_{2}≡ W(φ

_{s}, θ

_{s}), where φ

_{s}≡ φ

_{s}

_{2}, θ

_{s}≡ θ

_{s}

_{2}. The “weak” anchoring energy means such a value at which, due to the elastic interaction, the director at the boundary deviates markedly from the normal to the plane of the layer, so that the angle θ

_{s}can be close to π/2 and weakly dependent on q. Taking the first derivative of (1) by q, taking into account (2) and equating it to zero, we obtain a condition for the extreme of free energy:

_{s}= 0, the last term in (5) can be skipped, and it follows from (5) that if the tilt angle of director θ

_{s}at the boundary is different from π/2, then q/q

_{0}< 1 and, accordingly, the spiral pitch p in the CLC layer is greater than the natural pitch p

_{0}.

_{s}will approach π/2 and according to (5), the pitch p = 2π/q may decrease, approaching the natural pitch p

_{0}= 2π/q

_{0}(q/q

_{0}→ 1).

_{s}is an angle with respect to the x-axis coinciding with the direction of the in-plane electric field. Taking into account (2) at φ

_{s1}= 0, and φ

_{s}≡ φ

_{s2}, we obtain

_{0}. For clarity, when constructing the curves in Figure 3, the following parameter values were chosen: q

_{0}d = 10π, W

_{a}/(q

_{0}K

_{2}) = 0.5. For typical values K

_{2}= 5 10

^{−12}N and q

_{0}= 2π μm

^{−1}, as example, the last parameter corresponds to an anchoring energy magnitude of ~0.03 mJ/m

^{2}, which is within a reasonable range of values [27].

_{s}= π/2 and infinitesimal anchoring, the line f(ξ) crosses the ordinate axis at f(0) = −1 and the abscissa axis at ξ = q/q

_{0}= 1. Thus, in this case, the pitch of the spiral p is equal to the natural pitch p

_{0}. If there is a tilt of the director relative to the plane of the layer (θ

_{s}< π/2), then the line f(ξ) is raised along the ordinate axis by a value $\frac{\sqrt{{K}_{1}{K}_{3}}}{{q}_{0}d{K}_{2}}co{s}^{2}{\theta}_{s}$ (red curve), and the intersection point with the abscissa axis is shifted to lower values (ξ = q/q

_{0}< 1), which corresponds to an increased value of the pitch p compared to the natural pitch p

_{0}. In the latter case, the selective reflection band is shifted to the long-wave region relative to the natural-pitch band. If one increases the angle of tilt, for example, by using a localized electric field, then there will be a short-wave shift of the selective reflection band. This shift cannot exceed the value of the |(p

_{0}− p)(n

_{||}− n

_{^})/2|. In this case of zero azimuthal anchoring, the implemented spiral pitch corresponds to the global minimum of free energy.

_{s¬}lower than π/2, to the state “C”, when the tilt is increased to be close to π/2. Note that “D” is metastable, and, in principle, subsequent jump transitions are possible to the lower energy state "D" and further states up to the ground state at ξ = q/q

_{0}= 1. It is important, however, to emphasize that an increase in the strength of the electric field is equivalent to an increase in the amplitude of the azimuthal anchoring W

_{a}and, accordingly, the amplitude of the function g(ξ). The latter, in turn, will lead to increasing the energy barrier between the states “C” and “D”. It is this feature that explains the relatively small spectral shift to the short-wave region of the spectrum, observed experimentally, as seen in Figure 2a.

#### 3.3. Numerical Simulations and Discussion

_{1}, K

_{2}, and K

_{3}are splay, twist, and bend elastic coefficients, respectively; γ is rotational viscosity; ε

_{0}≅ 8.85 10

^{−12}F/m is the vacuum dielectric constant, and

**E**is the electric field vector; wavenumber q

_{0}is responsible for the LC chirality and spiral director distribution with the natural pitch p

_{0}= 2π/q

_{0}in case of the free boundary conditions and no electric field applied. The elastic parameters used in this work are as follows: K

_{1}= 11 pN, K

_{2}= 7 pN, K

_{3}= 17 pN, and γ = 0.1 Pa s, which are close to those of popular E7 LC from Merck. A value of q

_{0}is taken to be positive and defining the natural pitch of the right-handed helix p

_{0}= 0.5 μm. Namely, this value for the spiral pitch is used at the starting time moment of the calculations by the relaxation method. A value of 0.1 Pa s for the rotational viscosity (γ) is not of principal significance in current simulations; it defines only the relaxation time scale for obtaining a steady state solution. In our case, for the CLC layer of a thickness of 10 μm, more than one second was necessary to obtain a stable solution with an equilibrium spiral pitch p. To find the electric field, Equations (12) was coupled with the Maxwell equations $\nabla \times E=0$ and $\nabla \xb7\left({\epsilon}_{0}\epsilon E\right)=0$. Thus, we assumed that the LC is the ideal dielectric with zero free charge density in the bulk.

_{sx}and n

_{sy}are x- and y-components, respectively, of the LC director at the alignment surface. This is actually the well-known Papoular–Rapini form [32] for the anchoring potential, which, in our case, was expressed in terms of the director components with respect to the vertical easy axis direction.

_{||}= 1.81 and n

_{⊥}= 1.54 at the wavelength λ = 450 nm and n

_{||}= 1.75, and n

_{⊥}= 1.52 at the wavelength λ = 589 nm, so the spectral dispersion was taken into account. The spectra shown in this work are local and for the optical rays normal to the LC layer and passing through the middle point between the electrodes. The calculations were done for the nonpolarized light.

_{z}at the homeotropic-alignment surface. The field-off spectra (curves 1, 3) illustrated the well-known single selective reflection band in the near infrared range, which corresponds to the stop-band of width Δλ of about 100 nm, which is defined by pitch (p) of the spiral and the optical anisotropy (Δλ = p(n

_{||}− n

_{⊥})). One can see an impressively strong dependence of the stop-band position on the anchoring strength. Even at a rather weak anchoring strength (W

_{z}= 0.025 mJ/m

^{2}), the stop band is centered at a wavelength of λ

_{m}= 900 nm (curve 1), which is significantly red-shifted with respect to the wavelength λ

_{m}= p

_{0}( n

_{||}+ n

_{⊥})/2 ≅ 815 nm corresponding the CLC spiral characterized by the natural pitch p

_{0}= 500 nm. With increases in the zenithal anchoring strength to a value of 0.05 mJ/m

^{2}, a stop-band is further shifted to the longer wavelengths (λ

_{m}= 1090 nm), which corresponds to the spiral pitch of about 670 nm. The found increase in the spiral pitch with increasing anchoring strength agrees with our simplified analytical consideration.

_{0}[33]. This means that the zenithal anchoring strength in our experimental cells was rather weak and could be estimated at a level below 0.01 mJ/m

^{2}. Because of such weak zenithal anchoring provided by the alignment films, an important question arose about modification of the effective zenithal anchoring with the help of a localized electric field.

_{z}= 0.025 and W

_{z}= 0.05 mJ/m

^{2}respectively, as seen in Figure 4 (curves 2, 4). The simulations also reproduce the observed small shift of the main stop-band to the shorter wavelengths. In particular, the blue shift is found to be higher for the case of the higher anchoring strength (curve 4). We also found that the value of the shift depends on a thickness of the LC layer, which is actually in agreement with our analytical model (see the discussion with respect to Figure 3), when the in-plane electric field effectively changes the azimuthal anchoring, so a multiple set of metastable states appear.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic representation of two types of samples: (

**a**) the homeotropic alignment is implemented on the bottom substrate with planar electrodes, and planar alignment is on the top substrate (alignment of the easy axis is along the electrode strips); (

**b**) the planar alignment on the bottom substrate with electrodes, with the easy axis along the electrode strips; the top substrate provides the homeotropic alignment.

**Figure 2.**Transmittance spectra for samples “A” (

**a**) and “B” (

**b**) at different magnitudes of the driving voltage U

_{p-p}: 0, 100, 200, and 300 V for the curves 1, 2, 3, and 4, respectively.

**Figure 3.**Illustration of the graphic solution of Equation (8). A subset of the designated intersection points corresponds to a local minimum of free energy.

**Figure 4.**Optical transmittance spectra with the electric field switched off (curves 1, 3) and switched on (curves 2, 4) at voltage U = 100 V for two values of the zenithal anchoring energy on the homeotropic-alignment surface: W

_{z}= 0.025 mJ/m

^{2}(top); W

_{z}= 0.05 mJ/m

^{2}(bottom).

**Figure 5.**Equilibrium distribution of the LCD director near the homeotropic-alignment surface with weak anchoring (W

_{z}= 0.01 mJ/m

^{2}) for different values of voltage on interdigitated electrodes (electrode width 500 nm; gap between electrodes 200 nm). Images correspond to the xz plane (x-axis is horizontal direction, perpendicular to the electrode strips; the z-axis is vertical (normal to the plane of the CLC layer and corresponds to the spiral axis). The color scale corresponds to the x-component of the LC director.

**Figure 6.**Transmittance spectra of the CLC layer for different values of electrical voltage U on the interdigitated electrodes: (1) U = 0 V; (2) U = 2V; (3) U = 4V; (4) U = 8V; (5) U = 16V. The corresponding director distributions are shown in Figure 5.

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**MDPI and ACS Style**

Palto, S.P.; Rybakov, D.O.; Umanskii, B.A.; Shtykov, N.M.
Spiral Pitch Control in Cholesteric Liquid Crystal Layers with Hybrid Boundary Conditions. *Crystals* **2023**, *13*, 10.
https://doi.org/10.3390/cryst13010010

**AMA Style**

Palto SP, Rybakov DO, Umanskii BA, Shtykov NM.
Spiral Pitch Control in Cholesteric Liquid Crystal Layers with Hybrid Boundary Conditions. *Crystals*. 2023; 13(1):10.
https://doi.org/10.3390/cryst13010010

**Chicago/Turabian Style**

Palto, Serguei P., Dmitry O. Rybakov, Boris A. Umanskii, and Nikolay M. Shtykov.
2023. "Spiral Pitch Control in Cholesteric Liquid Crystal Layers with Hybrid Boundary Conditions" *Crystals* 13, no. 1: 10.
https://doi.org/10.3390/cryst13010010