Flexoelectricity and pattern formation in nematic liquid crystals

We present in this paper a detailed analysis of the flexoelectric instability of a planar nematic layer in the presence of an alternating electric field (frequency $\omega$), which leads to stripe patterns (flexodomains) in the plane of the layer. This equilibrium transition is governed by the free energy of the nematic, which describes the elasticity with respect to the orientational degrees of freedom supplemented by an electric part. Surprisingly the limit $\omega \to 0$ is highly singular. In distinct contrast to the dc case, where the patterns are stationary and time independent, they appear at finite, small $\omega$ periodically in time as sudden bursts. Flexodomains are in competition with the intensively studied electrohydrodynamic instability in nematics, which presents a nonequilibrium dissipative transition. It will be demonstrated that $\omega$ is a very convenient control parameter to tune between flexodomains and convection patterns, which are clearly distinguished by the orientation of their stripes.

Figure 1

Critical voltage $u_c$ (a) and critical wave number $p_c$ (b) as a function of $\mu$ for $\delta k=0$ (${\mu }_{\text{min}}=-1$, ${\mu }_{\text{max}}=1$), for $\delta k=0.3$ (${\mu }_{\text{min}}=-0.769$, ${\mu }_{\text{max}}=3.072$), and for $\delta k=-0.3$ (${\mu }_{\text{min}}=-1.427$, ${\mu }_{\text{max}}=0.413$).

Figure 2

Upper and lower limit curves, ${\mu }_{\text{max}}(\delta k)$ (dashed, red) and ${\mu }_{\text{min}}(\delta k)$ (solid, black), respectively, in the $(\delta k-\mu )$ plane. ${\mu }_{\text{max}}(\delta k)$ diverges for $\delta k\approx 0.53$, while ${\mu }_{\text{min}}\to -\infty$ for $\delta k\to -1$.

Figure 3

The neutral curve $u_0^2(p)$ normalized to the Freedericksz threshold $u_F$ Eq. (11) as a function of $p$ for $\delta k=0.2$ and different $\mu$ with ${\mu }_1=1.4,{\mu }_2=1.6,{\mu }_3=1.8,{\mu }_4=2.0$.

Figure 4

Upper limit curve ${\mu }_{\text{max}}$ (solid line, black) as a function $\delta k$ from Eq. (12) in comparison with ${\stackrel{̲}{\mu }}_{\text{max}}$ from Eq. (17) (dashed line, red).

Figure 5

Relative critical voltage $u_c^2/u_F^2$ and the critical wave number $p_c$ as function of $\delta k$ for $\mu =2$ (solid line for $u_c^2/u_F^2$ and dotted line for $p_c$, black) and in the absence of the flexoeffect ($e_1-e_3=0$) (dashed line for $u_c^2/u_F^2$ and dot-dashed line for $p_c$, red). For $\mu =2$ the Freedericksz state is approached at $\delta k_F\approx 0.206$, while this happens for $e_1-e_3=0$ at $\delta k=\delta k_{\text{ST}}$ [data from Eq. (12)].

Figure 6

The neutral curve $u_0^2(p)$ as a function of $p$ for $\delta k=-0.7$ and different $\mu$ where ${\mu }_1=0.11,{\mu }_2=0.12,{\mu }_3=0.13,\mathrm{and}\hspace{0.5em}{\mu }_4=0.14$.

Figure 7

Critical voltage $u_c(\omega )$ in dimensionless units (a) and critical wave number (b) of the flexodomains as function of $\omega {\tau }_d$ for MBBA parameters with the flexostrengths ${\xi }_{-}=2$ and ${\xi }_{-}=3$. The symmetry of the solution is indicated.

Figure 8

Critical voltage $u_c(\omega )$ in dimensionless units (a) and critical wave number (b) of the flexodomains as function of $\omega {\tau }_d$ for different $\delta k$ and ${\xi }_{-}=2$.

Figure 9

The director fields as a function of time at flexodomains threshold for a periodic excitation. $\omega {\tau }_d=0.05$, $\delta k=0.3$, and ${\xi }_{-}=2$ (conductive symmetry).

Figure 10

The critical effective voltage for flexodomains $u_c^W/\sqrt{2}$ (a) and the critical wave number $p_c^W$ (b) from Eq. (29) in the limit $\omega \to 0$ normalized to the dc values.

Figure 11

The director fields as function of time at the threshold of flexodomains for a square-wave excitation. $\omega {\tau }_d=0.05$, $\delta k=0$, and ${\xi }_{-}=2$ (dielectric symmetry).

Figure 12

Critical effective voltage $U_c$ of electroconvection (RMS value in volts) (a), modulus of the critical wave vector $\left|{\mathbf{q}}_c\right|$ (b), and the angle $\alpha$ of the critical wave vector with the $x$ axis (c) as a function of frequency $\omega$ (in units of the charge relaxation time ${\tau }_q$) for different magnitudes of the flexocoefficients $\xi ={\xi }_{-}={\xi }_{+}$ (for more details, see text).

Figure 13

The director fields as a function of time at EC threshold for $\omega {\tau }_q=2.3\times {10}^{-3}$ ($\omega {\tau }_d=0.1$).

Figure 14

Critical voltage $U_c$ (RMS value in volts) for EC as a function of frequency $\omega$ (in units of the charge relaxation time ${\tau }_q$) for five different values of ${\sigma }_a/{\sigma }_{\perp }$ between $0.5$ and $-0.5$ and for ${\xi }_{-}={\xi }_{+}=2$. The corresponding critical voltages for the flexodomains (conductive symmetry at small $\omega$ and dielectric symmetry at larger $\omega$) are included as well.

Figure 15

Critical effective voltage $U_c$ (RMS value in volts) as a function of frequency $\omega$ (in units of the charge relaxation time ${\tau }_q$) for flexodomains and EC patterns. Material parameters of MBBA except ${\sigma }_a/{\sigma }_{\perp }=0.2$, ${\xi }_{-}=3$, and ${\xi }_{+}=0.25$.

Figure 16

Critical effective voltage $U_c$ of electroconvection (RMS value in volts) as a function of frequency $\omega$ (in units of the charge relaxation time ${\tau }_q$) for MBBA material parameters (${\xi }_{+}={\xi }_{-}=1$) except ${\sigma }_a/{\sigma }_{\perp }=-0.05$.