Flexoelectricity and competition of time scales in electroconvection

An unexpected type of behavior in electroconvection (EC) has been detected in nematic liquid crystals (NLCs) under the condition of comparable time scales of the director relaxation and the period of the driving ac voltage. The studied NLCs exhibit standard EC (s-EC) at the onset of the instability, except one compound in which nonstandard EC (ns-EC) has been detected. In the relevant frequency region, the threshold voltage for conductive s-EC bends down considerably, while for dielectric s-EC it bends up strongly with the decrease of the driving frequency. We show that inclusion of the flexoelectric effect into the theoretical description of conductive s-EC leads to quantitative agreement, while for dielectric s-EC a qualitative agreement is achieved. The frequency dependence of the threshold voltage for ns-EC strongly resembles that of the dielectric s-EC.

Figure 1

(Color online) Frequency dependence of the s-EC threshold voltage $U_c$ measured in Phase 5 with $d=(3.4\pm 0.5)\mu \mathrm{m}$—conductive s-EC (circles) and dielectric s-EC (bullets). The solid and the dashed lines are the results of the numerical analysis with and without the flexoelectric effect, respectively. The inset is the magnification of the low frequency range. The arrow indicates the crossover frequency $f_c$.

Figure 2

(Color online) Numerical calculations for the frequency dependence of the s-EC threshold voltage $U_c$ in a $d=20\mu \mathrm{m}$ sample of Phase 5 with ${\sigma }_{\perp }=8.2\times {10}^{-8}{\left(\Omega \mathrm{m}\right)}^{-1}$. Dashed lines are the conductive s-EC and the dielectric s-EC solutions with $e_1=e_3=0$, while the solid lines are the solutions with $e_1=-26.5\mathrm{pC}∕\mathrm{m}$ and $e_3=-23.6\mathrm{pC}∕\mathrm{m}$. The inset is the magnification of the low frequency range. The arrow indicates the crossover frequency $f_c$.

Figure 3

(Color online) Numerically calculated frequency dependence of the s-EC threshold voltage $U_c$ for samples of various thicknesses in Phase 5 with ${\sigma }_{\perp }=8.2\times {10}^{-8}{\left(\Omega \mathrm{m}\right)}^{-1}$, $e_1=-26.5\mathrm{pC}∕\mathrm{m}$, and $e_3=-23.6\mathrm{pC}∕\mathrm{m}$. The circles and the star symbols on the curves denote $1∕{\tau }_d$ and the inflection point at $f_{\mathrm{infl}}$, respectively. Inset: the thickness dependence of $1∕f_{\mathrm{infl}}$.

Figure 4

(Color online) Frequency dependence of the threshold voltage $U_c$ in the conductive regime for a sample of Phase 5A with $d=3.1\mu \mathrm{m}$. Circles stand for experimental data. The solid line represents the $U_c\left(f\right)$ curve calculated numerically with Phase 5 parameters (the same as in Fig. 1, with nonzero $e_1$ and $e_3$) and with ${\sigma }_{\perp }=9.2\times {10}^{-8}{\left(\Omega \mathrm{m}\right)}^{-1}$. The dashed line is the calculated solution for $e_1=e_3=0$. The inset is the magnification of the low frequency range.

Figure 5

(Color online) Frequency dependence of the conductive s-EC threshold voltage $U_c$ measured on a sample of MBBA with $d=3.2\mu \mathrm{m}$ (circles). The solid line represents the $U_c\left(f\right)$ curve calculated numerically with MBBA parameters from Table and with ${\sigma }_{\perp }=9\times {10}^{-8}{\left(\Omega \mathrm{m}\right)}^{-1}$, $e_1=-14.5\mathrm{pC}∕\mathrm{m}$, and $e_3=-20.5\mathrm{pC}∕\mathrm{m}$. The dashed line is the solution for the same parameter set, but with $e_1=e_3=0$.

Figure 6

(Color online) Frequency dependence of the dielectric threshold voltage $U_c$ measured in a $d=3.4\mu \mathrm{m}$ sample of Phase 5 (bullets). Lines represent results of numerical calculations for Phase 5 with $e_1=-26.5\mathrm{pC}∕\mathrm{m}$, $e_3=-23.6\mathrm{pC}∕\mathrm{m}$, and ${\sigma }_{\perp }=1\times {10}^{-8}{\left(\Omega \mathrm{m}\right)}^{-1}$ for a series of ${\sigma }_a∕{\sigma }_{\perp }$ values as indicated.

Figure 7

(Color online) Numerical calculations of the threshold $U_c\left(f\right)$ for Phase 5 with ${\sigma }_a∕{\sigma }_{\perp }=0.1$, ${\sigma }_{\perp }=1\times {10}^{-8}{\left(\Omega \mathrm{m}\right)}^{-1}$, and $d=3.4\mu \mathrm{m}$: the dashed line is dielectric s-EC with $e_1=e_3=0$; the solid line is the dielectric mode with $e_1=-26.5\mathrm{pC}∕\mathrm{m}$, $e_3=-23.6\mathrm{pC}∕\mathrm{m}$.

Figure 8

(Color online) Frequency dependence of the dielectric s-EC threshold, calculated with Phase 5 parameters for ${\sigma }_a∕{\sigma }_{\perp }=0.1$ and with $e_1=-26.5\mathrm{pC}∕\mathrm{m}$, $e_3=-23.6\mathrm{pC}∕\mathrm{m}$, ${\sigma }_{\perp }=1\times {10}^{-8}{\left(\Omega \mathrm{m}\right)}^{-1}$ for different sample thicknesses $d$ as indicated. Circles denote $1∕{\tau }_d$ for each $d$. In the inset the thickness dependence of $1∕f_{\min }$ (bullets) and that of ${\tau }_d$ (solid line) is presented together with the best fit $1∕f_{\min }\propto d^2$ (dashed line).

Figure 9

(Color online) Results of numerical calculations for Phase 5 with ${\sigma }_a∕{\sigma }_{\perp }=0.1$, $e_1=-26.5\mathrm{pC}∕\mathrm{m}$, $e_3=-23.6\mathrm{pC}∕\mathrm{m}$, and $d=3.4\mu \mathrm{m}$ for three different conductivities ${\sigma }_{\perp }$.

Figure 10

(Color online) Frequency dependence of the s-EC threshold voltage $U_c$ measured in a sample of Phase 4 with $d=3.2\mu \mathrm{m}$ (circles). At dc voltage, the threshold for the flexoelectric domains is also indicated (solid triangle). For comparison, experimental data obtained on another sample of Phase 4 with $d=12.4\mu \mathrm{m}$ is also plotted (stars). The inset displays experimental results on the $d=12.4\mu \mathrm{m}$ cell in the whole frequency range studied.

Figure 11

Snapshots of patterns taken slightly above the threshold in a $d=3.2\mu \mathrm{m}$ sample of Phase 4. (a) Flexoelectric domains at dc voltage. (b) Coexisting flexoelectric domains and traveling dielectric oblique rolls at $f=15\mathrm{Hz}$. (c) Traveling dielectric oblique rolls at $f=70\mathrm{Hz}$. The arrow denotes the initial director. The physical size of the images is $(65\times 65)\mu {\mathrm{m}}^2$.

Figure 12

(Color online) Frequency dependence of the ns-EC threshold voltage $U_c$ measured in a sample of $\mathbf{8}∕\mathbf{7}$ with $d=3.4\mu \mathrm{m}$ for three different temperatures $T$. Solid lines represent linear fit to data at high frequencies (above the minima of the curves).

Figure 13

(Color online) ns-EC threshold curves $U_c\left(f\right)$ measured in samples of $\mathbf{8}∕\mathbf{7}$ of different thicknesses $d$ at $T=76°\mathrm{C}$. Solid lines represent linear fit to data.

Figure 14

(Color online) Frequency dependence of the threshold voltage $U_c$ measured in $\mathbf{8}∕\mathbf{7}$ samples of different thickness ($d=3.4\mu \mathrm{m}$, $13.2\mu \mathrm{m}$, and $40\mu \mathrm{m}$) at $T\approx 90°\mathrm{C}$ (in the vicinity of the s-EC to ns-EC transition). Open symbols denote the s-EC threshold, while closed symbols stand for ns-EC.

Figure 15

(Color online) Frequency dependence of the dimensionless critical wave number $q_c$ for Phase 5 with $d=(3.4\pm 0.5)\mu \mathrm{m}$. The solid and the dashed lines are the results of the numerical analysis with and without the flexoelectric effect, respectively. The inset is the magnification of the low frequency range. The arrow indicates the crossover frequency $f_c$.