Suppression of spatially periodic patterns by dc voltage

The effect of superposed dc and ac applied voltages on two types of spatially periodic instabilities in nematic liquid crystals, flexoelectric domains (FD), and electroconvection (EC) was studied. The onset characteristics, threshold voltages, and critical wave vectors were determined. We found that in general the superposition of driving with different time symmetries inhibits the pattern forming mechanisms for FD and EC as well. As a consequence, the onset extends to much higher voltages than the individual dc or ac thresholds. A dc-bias-induced reduction of the crossover frequency from the conductive to the dielectric EC regimes and a peculiar transition between two types of flexodomains with different wavelengths were detected. Direct measurements of the change of the electrical conductivity and its anisotropy, induced by the applied dc voltage component, showed that the dc bias substantially affects both parameters. Taking into account the experimentally detected variations of the conductivity in the linear stability analysis of the underlying nematohydrodynamic equations, a qualitative agreement with the experimental findings on the onset behavior of spatially periodic instabilities was obtained.

Figure 1

The chemical structure of the nematic 4-$n$-octyloxyphenyl 4-$n$-methyloxybenzoate (1OO8).

Figure 2

The frequency dependence of the ac threshold voltage $U_{\mathrm{c}}^{\mathrm{ac}}$ of EC in 1OO8 at various dc bias voltages $U_{\mathrm{dc}}$ in the frequency range of 5 Hz $\le f\le 300$ Hz. The vertical black arrow indicates the crossover frequency $f_{\mathrm{c}}$ between conductive (open symbols) and dielectric (solid symbols) EC regimes for $U_{\mathrm{dc}}=0$. At $U_{\mathrm{dc}}\ne 0$ only dielectric EC is observable.

Figure 3

Morphological phase diagram under combined dc and ac voltages with (a) $f=2$ Hz, (b) $f=5$ Hz, and (c) $f=10$ Hz. The abbreviations EC CR, EC DR, and FDSW correspond to conductive EC regime, dielectric EC regime, and flexodomains of short wavelength, respectively.

Figure 4

Typical pattern morphologies under combined dc and ac voltage with $f=5$ Hz: (a) flexodomains at $U_{\text{ac}}=1$ V, $U_{\text{dc}}=30$ V; (b) flexodomains at $U_{\text{ac}}=48$ V, $U_{\text{dc}}=80$ V; (c) oblique rolls in the conductive EC regime at $U_{\text{ac}}=7.4$ V, $U_{\text{dc}}=0$ V; (d) oblique rolls in the conductive EC regime at $U_{\text{ac}}=60$ V, $U_{\text{dc}}=72$ V. The double arrows show the direction of the initial director; their length corresponds to $100\mu \mathrm{m}$ ($d=19.5\mu \mathrm{m}$).

Figure 5

Two types of flexodomains under combined dc and ac voltage with $f=5$ Hz: (a) FD at $U_{\text{ac}}=12$ V, $U_{\text{dc}}=40$ V; (b) FDSW at $U_{\text{ac}}=55$ V, $U_{\text{dc}}=80$ V. Note the huge difference in the wavelength of the pattern. The double arrows show the initial director orientation; their length corresponds to $20\mu \mathrm{m}$ ($d=19.5\mu \mathrm{m}$).

Figure 6

Dependence of (a) the dimensionless wave number $q^{*}$, and (b) the obliqueness angle $\left|\alpha \right|$ of flexodomains and EC patterns on the ac voltage at $f=5$ Hz, close to threshold.

Figure 7

(a) Morphological phase diagram at superposing an ac voltage $U_{\mathrm{ac}1}$ of $f_1=400$ Hz (horizontal axis) with another ac voltage $U_{\mathrm{ac}2}$ of $f_2=10$ Hz ($U_2$, vertical axis). (b) The stability-limiting curves for the case when the ac voltage $U_{\mathrm{ac}2}$ of $f_2=10$ Hz is replaced by a dc voltage $U_{\mathrm{dc}}$.

Figure 8

Localized electroconvection structures and their development to extended pattern at the superposition of two ac voltages. The high-frequency component was kept constant, while the low $f$ component was increased from (a) around threshold to (b) 4% above threshold and (c) 8% above threshold. The double arrow shows the initial director orientation; its length corresponds to $400\mu \mathrm{m}$ ($d=50\mu \mathrm{m}$).

Figure 9

Phase diagram under combined dc and ac voltage with $f=2$ Hz exhibiting flexodomains of conductive type (FD cond) at dc driving and conductive EC regime (EC CR) at ac driving: (a) Stability-limiting curve in the $U_{\mathrm{ac}}-U_{\mathrm{dc}}$ plane, (b) the critical dimensionless wave numbers $q_{\mathrm{c}}^{*}$, and (c) the obliqueness angles $\alpha$ along the SLC.

Figure 10

Phase diagram under combined dc and ac voltage with $f=10$ Hz exhibiting flexodomains of conductive type (FD cond) at dc driving and dielectric EC regime (EC DR) at ac driving: (a) stability-limiting curve in the $U_{\mathrm{ac}}-U_{\mathrm{dc}}$ plane and (b) the critical dimensionless wave numbers $q_{\mathrm{c}}^{*}$ along the SLC.

Figure 11

Phase diagram under combined dc and ac voltage with $f=5$ Hz exhibiting transition between flexodomains of conductive type (FD cond) and of dielectric type (FD diel): (a) stability-limiting curve in the $U_{\mathrm{ac}}-U_{\mathrm{dc}}$ plane, (b) the critical dimensionless wave numbers $q_{\mathrm{c}}^{*}$, and (c) the obliqueness angles $\alpha$ along the SLC. For comparison, the SLC for the conductive EC regime (EC CR) is also shown with lower threshold voltages than for FD.

Figure 12

In the left panel, voltage dependence of (a) the parallel resistance $R_{\mathrm{p}}$, (b) the parallel capacitance $C_{\mathrm{p}}$ of a 1OO8 sample at $f=80$ Hz; (c) morphological phase diagram (voltage dependence of the contrast $\mathrm{\Psi }$) of the same sample. All 3D graphs are color coded: black corresponds to the lowest values, white to the highest values. In the right panel, 2D cross-sections of the 3D images (d) for $R_p$, (e) for $C_p$; (f) 2D projection of $\mathrm{\Psi }$. The dashed line corresponds to the stability-limiting curve of EC.

Figure 13

Dc voltage dependence of (a) the conductivities ${\sigma }_{\perp }$ and ${\sigma }_{\parallel }$, and (b) the relative conductivity anisotropy $\delta \sigma =({\sigma }_{\parallel }-{\sigma }_{\perp })/{\sigma }_{\perp }$. The solid (open) symbols plotted at the same $U_{\mathrm{dc}}$ correspond to data measured at the beginning (end) of the interval of about 8 min at which the dc voltage was kept constant.

Figure 14

Temporal variation of (a) the conductivity ${\sigma }_{\perp }$ and ${\sigma }_{\parallel }$, and (b) the relative conductivity anisotropy $\delta \sigma ={\sigma }_{\mathrm{a}}/{\sigma }_{\perp }$ during steplike increase of the dc voltage from $U_{\mathrm{dc}}=0$ to $U_{\mathrm{dc}}=10$ V, and then from $U_{\mathrm{dc}}=10$ V to $U_{\mathrm{dc}}=0$. The vertical dashed lines mark the moments of the voltage jumps. The solid lines correspond to fits by a superposition of two exponential decay functions according to Eq. (1).

Figure 15

Calculated frequency dependence of (a) the threshold voltages $U_{\mathrm{c}}^{\mathrm{ac}}$, (b) the dimensionless critical wave number $q_{\mathrm{c}}^{*}$, and (c) the obliqueness angle $\alpha$ of EC patterns at pure ac driving at the conductivity anisotropy of ${\sigma }_{\mathrm{a}}/{\sigma }_{\perp }=0.5$ and different values of the conductivity ${\sigma }_{\perp }$.

Figure 16

Calculated frequency dependence of (a) the threshold voltages $U_{\mathrm{c}}^{\mathrm{ac}}$, (b) the dimensionless critical wave number $q_{\mathrm{c}}^{*}$, and (c) the obliqueness angle $\alpha$ of EC patterns at pure ac driving at the conductivity of ${\sigma }_{\perp }=4\mathrm{nS}{\mathrm{m}}^{-1}$ and different values of the conductivity anisotropy ${\sigma }_{\mathrm{a}}/{\sigma }_{\perp }$.

Figure 17

The stability-limiting curves under combined dc and ac voltages in the $U_{\mathrm{ac}}-U_{\mathrm{dc}}$ plane for ${\sigma }_{\mathrm{a}}/{\sigma }_{\perp }=0.5$ and different conductivity values ${\sigma }_{\perp }$, calculated with 1OO8 parameter set for (a) the conductive regime ($f=5$ Hz), and (b) the dielectric regime ($f=25$ Hz). Diamonds were the thresholds, if one assumes ${\sigma }_{\perp }=4\mathrm{nS}{\mathrm{m}}^{-1}$ at $U_{\mathrm{dc}}=0, {\sigma }_{\perp }=3\mathrm{nS}{\mathrm{m}}^{-1}$ at $U_{\mathrm{dc}}=2$ V, and ${\sigma }_{\perp }=2\mathrm{nS}{\mathrm{m}}^{-1}$ at $U_{\mathrm{dc}}=4$ V. The dash-dotted line indicates the trend of the resulting stability-limiting curve.

Figure 18

The stability-limiting curves under combined dc and ac voltages in the $U_{\mathrm{ac}}-U_{\mathrm{dc}}$ plane for ${\sigma }_{\perp }=4\mathrm{nS}{\mathrm{m}}^{-1}$ and different conductivity anisotropy values ${\sigma }_{\mathrm{a}}/{\sigma }_{\perp }$, calculated with 1OO8 parameter set for (a) the conductive regime ($f=5$ Hz), and (b) the dielectric regime ($f=25$ Hz). Diamonds were the thresholds, if one assumes ${\sigma }_{\mathrm{a}}/{\sigma }_{\perp }=0.5$ at $U_{\mathrm{dc}}=0, {\sigma }_{\mathrm{a}}/{\sigma }_{\perp }=0.4$ at $U_{\mathrm{dc}}=2$ V and ${\sigma }_{\mathrm{a}}/{\sigma }_{\perp }=0.3$ at $U_{\mathrm{dc}}=4$ V. The dash-dotted line indicates the trend of the resulting stability-limiting curve.