Nonstandard electroconvection and flexoelectricity in nematic liquid crystals

For many years it has been commonly accepted that electroconvection (EC) as primary instability in nematic liquid crystals for the “classical” planar geometry requires a positive anisotropy of the electric conductivity, ${\sigma }_a$, and a slightly negative dielectric anisotropy, ${ϵ}_a$. This firm belief was supported by many experimental and theoretical studies. Recent experiments, which have surprisingly revealed EC patterns at negative conduction anisotropy as well, have motivated the theoretical studies in this paper. It will be demonstrated that extending the common hydrodynamic description of nematics by the usually neglected flexoelectric effect allows for a simple explanation of EC in the “nonstandard” case ${\sigma }_a<0$.

Figure 1

(Color online) Critical voltage $U_c$ as function of $\omega {\tau }_q$ for different ${\sigma }_a∕{\sigma }_{\perp }$ as indicated by the arrows.

Figure 2

(Color online) Modulus of the critical wave vector ${\mathit{q}}_c$ as function of $\omega {\tau }_q$ for different ${\sigma }_a∕{\sigma }_{\perp }$.

Figure 3

(Color online) Angle $\alpha$ of the critical wave vector ${\mathit{q}}_c$ with the $x$ axis as function of $\omega {\tau }_q$ for different ${\sigma }_a∕{\sigma }_{\perp }$.

Figure 4

Temporal evolution of the director amplitude $n_z(0,t)$ for one period $0<\omega t<2\pi$ of the driving ac voltage.

Figure 5

(Color online) Profile of the amplitude $n_z(z,t)$ along $z$ for different times. The dashed curve marked as “avr.” represents the time average.

Figure 6

(Color online) Profile of the amplitude ${\overline{\rho }}_{fl}(z,t)$ of the flexo charge density along $z$ for different times.

Figure 7

(Color online) Profile of the amplitude of the Coulomb charge density ${\overline{\rho }}_C={\overline{\rho }}_{el}-{\overline{\rho }}_{fl}$ along $z$ for different times.

Figure 8

(Color online) Profile of the torque density ${\Gamma }_y(z,t)$ on the director along $z$ for different times.

Figure 9

(Color online) Neutral curves $U_0\left(q\right)$ for the wave vector $\mathit{q}=(q,p)$ as a function of $q$ at a fixed value of $p$ for different magnitudes of the flexo coefficients $({\xi }_{-},{\xi }_{+})$ at $\omega {\tau }_q=0.5$.

Figure 10

(Color online) Experimental data (symbols) and theoretical results (lines) for the critical voltage $U_c$ as a function of frequency $f$ for the compound $\mathbf{8}∕\mathbf{7}$ for positive as well as for negative ${\sigma }_a$.

Figure 11

(Color online) Modulus of the critical wave vector ${\mathit{q}}_c$ as a function of frequency $f$ for the compound $\mathbf{8}∕\mathbf{7}$ for positive as well as for negative ${\sigma }_a$. Experimental data (symbols) and theoretical results (lines).