Steady state-Hopf mode interactions at the onset of electroconvection in the nematic liquid crystal Phase V.

We report on a new mode interaction found in electroconvection experiments on the nematic liquid crystal mixture Phase V in planar geometry. The mode interaction (codimension two) point occurs at a critical value of the frequency of the driving AC voltage. For frequencies below this value the primary pattern-forming instability at the onset voltage is an oblique stationary instability involving oblique rolls, and above this value it is an oscillatory instability giving rise to normal traveling rolls (oriented perpendicular to and traveling in the director direction). The transition has been confirmed by measuring the roll angle and the dominant frequency of the time series, as both quantities exhibit a discontinuous jump across zero when the AC frequency is varied near threshold. The globally coupled system of Ginzburg-Landau equations that qualitatively describe this mode interaction is constructed, and the resulting normal form, in which slow spatial variations of the mode amplitudes are ignored, is analyzed. This analysis shows that the Ginzburg-Landau system provides the adequate theoretical description for the experimentally observed phenomenon. The experimentally observed patterns at and higher above the onset allow us to narrow down the range of the parameters in the normal form.

1. Introduction

Electroconvection in nematic liquid crystals is a physical system that serves as a testing ground for experimental studies and theoretical predictions of pattern formation in spatially extended anisotropic systems [1-5]. Its advantages over other pattern forming systems are the high aspect ratio (~103), short time scales, and the adjustable control parameters (amplitude and frequency of the applied voltage). For electroconvection, the nematic is sandwiched between two glass plates, and an AC voltage is applied across the electrode plates. Above a critical value of the applied voltage, an electrohydrodynamic instability combined with a transition from the uniform state to a variety of patterns can occur, including stationary and traveling rolls as well as more complex spatiotemporal structures like worms, defects and spatiotemporal chaos [1,6-10].

Some of the patterns observed near the onset can be described by the standard model [5,11], which combines the theory of Ericksen and Leslie for an electrically conducting anisotropic fluid with the quasistatic Maxwell equations under the assumption of an ohmic charge conduction in the liquid crystal. Even with extra flexoelectric term, the standard model always predicts a stationary instability leading to stationary rolls, so it does not exhibit any oscillatory instability giving rise to the traveling wave patterns frequently observed near the onset in a variety of nematics like MBBA and Phase V [7,12,13], and I52 [14]. The weak electrolyte model [15], an extension of the standard model, in which a slow dissociation-recombination process of the charge carrying ions is taken into account and the ohmic behavior is replaced by the dynamics of two species of oppositely charged mobile ions, provides a basis for understanding the Hopf bifurcation that predicts the traveling wave patterns observed experimentally [7,16]. The weak electrolyte model can show an oscillatory instability as well as a stationary instability at the onset, and both types of instabilities can lead to oblique or normal rolls depending on the parameters, thus allowing for the occurrence of steady-oscillatory mode interactions.

A number of experimental and analytical studies of steady-oscillatory (Hopf) mode interactions in isotropic pattern forming systems, like Taylor–Couette flow, Rayleigh–Bénard convection, and convection in binary mixtures have been reported in the literature (see, e.g., [17-21] and references therein). There are no similar results in the literature for steady state-Hopf mode interactions in anisotropic systems. In this paper we report and analyze complex spatiotemporal patterns recorded in the electroconvection of the nematic mixture Phase V for AC voltages slightly above the onset value, which are dominated by normal oscillatory/oblique steady-state mode interaction. We observed, for two cells of almost the same thickness, oblique stationary modes at lower frequencies and normal traveling modes at higher frequencies, as well as a jump in the Hopf frequency from zero (stationary state) to over 20 rad/sec (normal traveling rolls) as the driving frequency is increased above a critical value.

The mode interaction point is identified with this critical value of the AC frequency along with the associated electroconvective threshold voltage. Since two parameters are fixed, this mode interaction point marks a codimension-two bifurcation point, which is rather different from the well known Lifshitz point. At the Lifshitz point a continuous transition from normal to oblique rolls of the same type (either stationary of traveling) occurs, whereas the codimension-two point reported in this paper marks a transition from oblique stationary to normal traveling rolls combined with a discontinuity in the Hopf frequency as well as wave numbers and roll angle.

The experimentally observed behavior can be understood through the amplitude equations that follow from a governing system of partial differential equations for anisotropic systems in a weakly nonlinear analysis. In this paper we do not pursue this analysis for the equations for nematic electroconvection, but instead use symmetry considerations to set up the generic form of the system of globally coupled Ginzburg–Landau equations that describes the patterns observable near the mode interaction point. The analysis of the resulting normal form, in which slow variations of the amplitudes are ignored, in a range of parameters consistent with the experiments, provides a possible theoretical scenario corresponding to the experimentally observed phenomena.

We report here, for the first time in the literature, experimental evidence of a steady oblique–normal traveling mode interaction point. Our experiments have been performed in the conduction regime, below the cutoff value of the driving AC frequency. In a previous study [22], a phase diagram for a Phase V sample sandwiched in a “channel” (thin cell in one direction) is presented in which a transition from traveling to steady convection occurs at the cutoff frequency. This transition also corresponds to a steady-traveling mode interaction, however it does not involve interaction of rolls oriented in different directions. Moreover, due to the small aspect ratio of the channel, the amplitude description involves 1D Ginzburg–Landau equations, and both traveling and steady rolls are normal rolls (of different wavelengths). In contrast, the amplitude equations needed to capture our experimental situation are 2D anisotropic Ginzburg–Landau equations.

The outline of the paper is as follows. In Section 2 we describe the experimental setup and the observed electroconvective patterns. In Section 3 we introduce a system of globally coupled Ginzburg–Landau amplitude equations that captures near-onset patterns in the neighborhood of the mode interaction point, and perform a bifurcation analysis of the associated normal form. Section 4 is devoted to a discussion of the results and concluding remarks.

2. Experiments

2.1. Experimental Set Up

The experiments were conducted with the nematic liquid crystal (NLC) Phase V [7,23], a mixture of 65 wt% 4-butyl-4′-methoxyazoxybenzene and 35 wt% 4-ethyl-4′-methoxyazoxybenzene with nematic range between −5 °C and 75 °C. The NLC was sandwiched in a ready made sample cell fabricated by E.H.C. Co., Tokyo, Japan (EHC-cell) with flat transparent electrodes, which are rubbed to produce planar alignment of the director in a fixed (x) direction. The electrodes are made of indium tin oxide coated borosilicate glass plates and provide an active lateral area of A = 10 × 10mm2. Outside of the active area, there is no conductive coating and hence no electrical field is present. In this parallel-plate-capacitor geometry, the electrodes are separated by a vertical distance, d, which was measured interferometrically. The electrical contact between the plates and the hookup wires was created using silver-laden epoxy.

Usually a small amount of dopant (tetrabutylammium bromide) is added to the NLC to increase the conductivity. For our sample the conductivity was sufficient to observe the desired electroconvection states, thus the experiments have been conducted without added dopant. The conductivity varies between individual cells with temperature, AC frequency, and time elapsed, and threshold voltages vary accordingly. To obtain the conductivity, σ⊥, and the dielectric permittivity, ε⊥, perpendicular to the director, we measured the conductance (G⊥ = Aσ⊥/d) and the capacitance (C⊥ = Aε0ε⊥/d) using a capacitance bridge.

The electroconvection apparatus consisted of a temperature-controlled hot stage, electronics for applying the AC voltage, and a shadowgraph [24] apparatus for visualization. The cell was illuminated by polarized light with the polarization along the director, and the resulting shadowgraph images were monitored by a charge-coupled device camera mounted on the microscope, using a 10× objective. The frame grabber hooked up to the camera captures 8 bit grey scale images of size 480 × 640 pixels at a rate of up to 30 frames per second. The images recorded cover an area of 358.21 × 477.61 μm2 from the active area of the cell. In order to remove inhomogeneities in the optical system, the raw images have been flat-field corrected using the background and dark frames of the system, see [25] for details.

The EHC experiments have been performed in two cells A and B of thicknesses

The experiments in cell B were control experiments to confirm the reproducibility of the EHC-patterns observed in cell A. The electroconvection was driven by applying an AC voltage of frequency f0 (circular frequency ω0) and voltage V to the electrodes. The driving frequency was increased in certain steps adapted to the variation of the onset voltage V. For our sample the dielectric anisotropy was negative, and the AC frequency was varied in the conduction regime, i.e., below the cutoff frequency which was calculated from measured and tabulated [7] material parameters using the formula given in [11].

To record near-onset patterns for a fixed AC frequency, we slowly increased V (0 < ε = V2/V2c – 1 ≪ 1), waited a few minutes, and then captured a shadowgraph image and calculated its spatial power spectrum S(k), k = (p, q). Focusing was adjusted to enhance the dominant inner modes (first harmonics) of S(k). The sample stage was rotated to ensure that the peaks of S(k) were equidistant from the axes. Next, the stage was fixed for the whole experiment, and a short movie of 2048 frames was captured. To see changes in the patterns above the onset, for some frequencies the experiments have been repeated for higher values of V.

To characterize the temporal behavior of the near-onset patterns, we computed the power spectrum, P(ω), of the time series of the central pixel value for each individual movie. A sharp maximum of P(ω)at a nonzero circular frequency ω > 0 was considered as the indicator of traveling rolls, and the instability at the onset was identified with a Hopf bifurcation (HB) (Hopf frequency f = ω/2π). An example of P(ω) with ω > 0 is shown in Figure 1. If the maximum occurred at ω ≈ 0, we identified the instability with a steady state bifurcation (SSB). In our experiments normal traveling (NT) rolls were excited at Hopf bifurcations, and oblique stationary (OS) rolls at steady state bifurcations.

Figure 1

P(ω) for cell A at 35 °C and f0 = 350 Hz corresponding to ω0τq = 1.07, where τq is the charge relaxation time. The circular Hopf frequency is ω = 16.21 radians/s (f = 2.58 Hz).

2.2. Results

Figure 2(a) shows the onset voltage for cell A at 35 °C as function of the dimensionless driving frequency ω0τ, where τ = ε0ε⊥/σ ⊥ is the charge relaxation time. The variation of ω in this range is displayed in Figure 2(b). While the threshold curve is continuous, we observe two discontinuous jumps in ω which are marked by vertical dashed lines. The first jump is from ω = 0 to ω = 12.57 radians/s (f = 2.0 Hz) and occurs at f0 = 92 Hz (ω0τ = 0.28). This jump clearly corresponds to a transition from a steady state bifurcation to a Hopf bifurcation (Section 3). The origin of the second jump is unknown at present.

Figure 2

(a) Threshold curve V and (b) circular frequency ω versus ω0τ for cell A at 35 °C. In (c) ω is shown for cell B at 35 °C and 40 °C. The vertical dashed lines in (b) and (c) mark discontinuous jumps.

To test the reproducibility of the discontinuity in ω, we have performed the same experiment in cell B at 35 °C and 40 °C. The variations of ω with the driving frequency for these temperatures are shown in Figure 2(c). In both cases we can recognize again the jumps from ω = 0 to nonzero value of ω occurring at values of ω0τ close to 0.28. The second discontinuity found in cell A is either not present or occurs at a higher value, outside of the frequency range studied for cell B, i.e., at some ω0τ > 1.

In order to characterize the transition at the first jump, we studied the patterns created slightly above the onset for ω0τ below and above the discontinuity in ω. In Figure 3(a),(b), near-onset pattern snapshots are shown for f0 = 90 Hz (ω0τ = 0.27) and f0 = 95 Hz (ω0τ = 0.29), with voltages V = 10.38 V (ε = 0.05) and V = 10.39 V (ε = 0.016), respectively. The pattern in Figure 3(a) is stationary (which can be seen in the movie) and made up of oblique rolls, whereas the almost normal rolls apparent in (b) are traveling vertically upwards. The moduli of the Fourier transforms of the images shown in Figure 3(a),(b) are shown in Figure 4(a),(b). For both patterns, we can clearly recognize sharp peaks at the dominant (critical) wave numbers corresponding to oblique and normal rolls. The peaks at higher harmonics (not shown in Figure 4) are significantly smaller than the peaks at the first harmonics.

Figure 3

Snapshots of patterns in cell A at 35 °C for (a) f0 =90 Hz (ω0τ =0.27), V =10.38V (ε = 0.05), and (b) f0 =95 Hz (ω0τ =0.29), V =10.39V (ε = 0.016).

Figure 4

(a) and (b): Amplitude of Fourier transform of the images depicted in Figure 3(a), (b), respectively, into regions showing the dominant first harmonics.

When the applied voltage in cell A at 35 °C at f0 = 95 Hz was increased further above threshold, we observed a stationary pattern made up of oblique rolls again (not shown here). In contrast, when V was further increased above threshold for f0 = 90 Hz, no qualitative change in the pattern dynamics was observed. Thus above the jump the near-onset patterns are NT rolls, but the OS rolls reappear when the voltage is increased further.

To confirm the transition from normal to oblique rolls at onset, we computed the average of the horizontal and vertical wave numbers,

where S(p, q) is the average of S(p, q) over the first 100 frames, normalized such that ∫S(p, q)dpdq = 1. The variations of these averages, and of the corresponding average roll angle,

with ω0τ are shown in Figure 5(a, b), respectively. In these plots we can see jumps in and θ at ω0τ ≈ 0.28 from nonzero to zero values, showing that there is a transition from oblique to normal rolls. Notice that in the oblique regime