Twist-Bend Nematic Phase: Role of Third-Order Legendre Polynomial Term in Chiral Interaction Potential

Abstract

Using a mean-field theory, a twist-bend nematic (N_TB) phase has been described. In addition to second-order Legendre polynomial term, the chiral interaction potential includes third-order Legendre polynomial term also. It considers the coupling between a nematic director and a pitch axis of the N_TB phase. The distortion free energy, order parameter for twist-bend nematic, and orientational distribution function in equilibrium state have been calculated. Based on the free energy minimization, we obtain that the inclusion of third-order Legendre polynomial term in interaction energy makes N_TB phase more stable.

Appendix: Determination of Distortion Free Energy (Fdis)

In this appendix, we calculate the distortion free energy for higher order Legendre polynomial in chiral interaction. The anisotropic free energy is given by F_ani = F_nem + F_d. The distortion free energy is given as

$$ \frac{a^3\beta {F}_d}{V}=\frac{1}{4}{\nu}_L\frac{2}{9}{d}_0^2{\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right){\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right)-\kern0.5em \frac{2}{9}{c}_L{d}_0{\varepsilon}_{\alpha \beta \gamma}{Q}_{\mu \beta}\left(\boldsymbol{r}\right){\partial}_{\alpha }{Q}_{\mu \gamma}\left(\boldsymbol{r}\right) $$

(27)

where ∂_훾 = ∂/∂r is the first spatial derivative of the tensor order parameter and ε_αβγ is the Levi-Civita anti-symmetric tensor of the third rank. By using the tensor order parameter, we get

$$ {\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right){\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right)=\frac{9}{2}{S}_L^2{\partial}_{\alpha }{n}_{\beta }{\partial}_{\alpha }{n}_{\beta } $$

(28)

We also have

$$ {\partial}_{\alpha }{n}_{\beta }{\partial}_{\alpha }{n}_{\beta }={\overset{\sim }{k}}_1{\left(\nabla \cdotp \boldsymbol{n}\right)}^2+{\overset{\sim }{k}}_2{\left(n\cdotp \nabla X\ n\right)}^2+{\overset{\sim }{k}}_3{\left(n\ X\nabla X\ n\right)}^2, $$

(29)

where contribution to the free energy due to surface is neglected and the coefficient k_i(i = 1, 2, 3) is a dimensionless elastic constant, normalized by the typical value (~ 10⁻⁶ dyn).

Of the elastic constant of a pure material, 휙_L = 1 with S_L = 1.

$$ {\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right){\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right)=\frac{9}{2}{S}_L^2\left[\ {\overset{\sim }{k}}_1{\left(\nabla \cdotp \boldsymbol{n}\right)}^2+{\overset{\sim }{k}}_2{\left(n\cdotp \nabla X\ n\right)}^2+{\overset{\sim }{k}}_3{\left(n\ X\nabla X\ n\right)}^2\right] $$

(30)

Using the director n for the N_TB phase, we have

$$ {\left(\nabla \cdotp \boldsymbol{n}\right)}^2=0 $$

(31)

$$ {\left(n\cdotp \nabla X\ n\right)}^2={\left({\partial}_z\omega \right)}^2={q}^2{\sin}^4\varepsilon \kern0.5em $$

(32)

And

$$ {\displaystyle \begin{array}{c}{\left(n\ X\nabla X\ n\right)}^2={\left({\partial}_z\omega \right)}^2{\sin}^2\varepsilon\ \left(1-{\sin}^2\varepsilon \right)\\ {}={q}^2{\sin}^2\varepsilon \left(1-{\sin}^2\varepsilon \right)\end{array}} $$

(33)

Substituting Eq. (31), (32), and (33) into Eq. (30), we get

$$ {v}_L{\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right){\partial}_{\gamma }{Q}_{\alpha \beta}\left(\boldsymbol{r}\right)=\frac{9}{2}{S}_L^2{q}^2{\sin}^2\varepsilon \left[{\overset{\sim }{k}}_2{\sin}^2\varepsilon +{\overset{\sim }{k}}_3\left(1-{\sin}^2\varepsilon \right)\right] $$

(34)

In Eq. (34), the first term is for twist distortion and second term stands for bend distortion.

In Eq. (27), distortion free energy term is

$$ {c}_L{\varepsilon}_{\alpha \beta \gamma}{Q}_{\mu \beta}\left(\boldsymbol{r}\right){\partial}_{\alpha }{Q}_{\mu \gamma}\left(\boldsymbol{r}\right)={c}_L\frac{9}{4}{\overset{\sim }{k}}_2{S}_L^2q{\sin}^2\varepsilon $$

(35)

In above equation, dimensionless elastic constant k₂is because of the twist term.

Substituting Eqs. (34) and (35) into Eq. (27), we get

$$ \frac{a^3\beta {F}_d}{V}=\frac{1}{2}{v}_L{S}_L^2\ \left[\frac{1}{2}\left({\overset{\sim }{k}}_2-{\overset{\sim }{k}}_3\right){Q}^2{y}^2+\right(\frac{1}{2}{\overset{\sim }{k}}_3Q-{\overset{\sim }{k}}_2{Q}_0\left(\ {Q}_y\right] $$

(36)

where order parameter is y = sin²ε and the pitch wavenumber is represented by Q = qd₀. For N* phase, the dimensionless wavenumber is defined by Q₀ = q₀d_0.

From elastic continuum theory, ${\overset{\sim }{k}}_{11}=0$ and

$$ {\overset{\sim }{k}}_{22}=\frac{1}{2}{k}_BT\frac{S_L^2{v}_L{\overset{\sim }{k}}_2}{a} $$

(37)

and for twist elastic constant

$$ {\overset{\sim }{k}}_{33}=\frac{1}{2}{k}_BT\frac{S_L^2{v}_L{\overset{\sim }{k}}_3}{a} $$

(38)

The ratio of both the elastic constant is a constant.

The coupling free energy from Eq. (12) is written as

$$ \frac{a^3\beta {F}_{co}}{V}=-{s}_L{\upgamma}_{\mathrm{L}}\left(1-\frac{3}{2}\ y\right) $$

(39)