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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.

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 [^{3}), 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 [

Some of the patterns observed near the onset can be described by the standard model [

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., [

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

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 [

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.

The experiments were conducted with the nematic liquid crystal (NLC) Phase V [^{2}. 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,

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, _{0}

The electroconvection apparatus consisted of a temperature-controlled hot stage, electronics for applying the AC voltage, and a shadowgraph [^{2} 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 [

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 _{0} (circular frequency _{0}) and voltage _{c}

To record near-onset patterns for a fixed AC frequency, we slowly increased ^{2}/^{2}_{c} – 1 ≪ 1), waited a few minutes, and then captured a shadowgraph image and calculated its spatial power spectrum

To characterize the temporal behavior of the near-onset patterns, we computed the power spectrum, _{H}_{H}_{H}_{H}_{H}

_{0}_{q}_{q}_{0}_{H}_{H}_{H}_{H}_{H}_{0} = 92 Hz (_{0}_{q}

To test the reproducibility of the discontinuity in _{H}_{H}_{H}_{H}_{0}_{q}_{0}_{q}

In order to characterize the transition at the first jump, we studied the patterns created slightly above the onset for _{0}_{q}_{H}_{0} = 90 Hz (_{0}_{q}_{0} = 95 Hz (_{0}_{q}

When the applied voltage in cell A at 35 °C at _{0} = 95 Hz was increased further above threshold, we observed a stationary pattern made up of oblique rolls again (not shown here). In contrast, when _{0} = 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 _{a}_{a}

with _{0}_{q}_{0}_{q}_{⊥} and _{⊥} (but not a discontinuity) in the oblique regime, see

The observed coincidence of a stationary and an oscillatory instability is commonly referred to as a codimension-two mode interaction [^{2}^{2})-space, on which an eigenvalue of the linearized system is either zero (stationary case) or purely imaginary (oscillatory case).

Let us denote by _{s}^{2}^{2}_{0}) the stationary neutral stability surface with minimum _{sc}_{0}) = _{s}^{2}_{sc}^{2}_{sc}_{0}), and by _{o}^{2}^{2}_{0}) the oscillatory neutral stability surface with minimum _{oc}_{0}) = _{o}^{2}_{oc}^{2}_{oc}_{0}) and Hopf frequency _{H}_{0}) at criticality. Our experiments suggest that _{oc}_{0}) = 0 since NT rolls are observed, and _{oc}_{0}) = _{sc}_{0}) at the critical value of the AC frequency (_{0}_{c}_{q}_{oc}_{0}) _{sc}_{0}) if _{0} _{0}_{c}_{oc}_{0}) _{sc}_{0}) if _{0} _{0}_{c}_{sc}_{0})_{sc}_{0})) for _{0} _{0}_{c}_{oc}_{0})_{0} _{0}_{c}_{0} = _{0}_{c}_{c}_{c}_{sc}_{0}_{c}_{oc}_{0}), and has frequency _{Hc}_{H}_{0}_{c}

In general there should be no special relation between the location of the minima on the two neutral stability surfaces, thus a jump can be expected in both the horizontal and vertical critical wave numbers as we have found in the experiments. We note that the codimension-two mode interaction point described above is very different from the well-known Lifshitz point [

We now describe the derivation of the system of globally coupled Ginzburg–Landau amplitude equations that captures near-onset patterns in a vicinity of the mode interaction point.

The weak electrolyte model (WEM) consists of partial differential equations derived from the Navier-Stokes equation for an anisotropic electrically conducting fluid, the conservation of charge, Poisson’s law, and a partial differential equation for the conductivity. The WEM equations are extremely complicated for a fully 3D numerical simulation, therefore a weakly nonlinear analysis at the onset is particularly useful. In this analysis, the patterns above threshold are represented as superposition of OS and NT modes in the form

where u represents the field variables of the WEM (velocities, electric potential, director, conductivity, see [^{2} ~ |_{c}_{s±}_{o±}^{2}_{+}_{−}_{±} = _{c}_{c}

The form of the system of globally coupled Ginzburg–Landau equations for the envelopes follows in a straightforward manner from symmetry considerations combined with a formal multiple scale expansion and an appropriate rescaling of the envelopes and the slow variables as (the subscript

where _{1}, _{2},

with a further real coefficient ^{2} _{c}_{1}, _{2},

where _{+} if _{−} if _{s}_{o}_{o}

where

with a further real coefficient _{s}_{o}_{o}_{s}

We note that the system (_{sc}_{oc}

The Ginzburg–Landau system (

In polar coordinates, _{A}^{iϕA}

The basic (conduction) state corresponds to the trivial solution T : _{s}_{or}_{s}_{A}_{B}_{C}_{D}_{A}_{B}_{C}_{D}_{s}_{or}

At Λ_{or}_{C}_{D}_{A}_{B}_{C}_{D}_{A}_{B}

As common in mode interaction normal forms [_{A}_{o}_{C}_{n}_{B}_{D}

The equations for OS, NT, and MM are

where

which we assume to be nonzero. The MM solution is a quasiperiodic solution of (

The OS-solution exists in Λ_{s}

Likewise the NT-solution exists in Λ_{or}

The two half-lines (O) and (N) define a wedge in the (Λ_{s}_{or}

In _{s}_{or}_{r}_{r}_{0} = _{0}_{c}_{c}_{c}_{r}_{r}

The path for

The bifurcation diagram

Which implies that the path in _{1}_{r}_{or}

Stability exchanges of the primary branches occur when the perturbed path for _{0} _{0}_{c}_{oc}

In summary, consistency with our experiments requires that the conditions (_{0} _{0}_{c}

In the normal form description, the critical voltages are given by Λ_{s}_{or}

The threshold curves depicted in _{sc}_{c}_{oc}_{c}_{s}_{or}

In this paper we have presented and analyzed the first reported occurrence of near-onset patterns dominated by the interaction of steady oblique rolls and normal traveling rolls in nematic elctroconvection experiments. The results described in this paper confirm that nematic electroconvection is a multi-parameter physical system that naturally exhibits this kind of mode interaction. In addition, our experiments also confirm the weak electrolyte model as the correct theoretical description governing the spatiotemporal dynamics of nematic elctroconvection, since it predicts oblique as well as normal rolls at the onset and both types of rolls can be stationary or traveling. As common in spatially extended systems, we did not observe ideal roll patterns, but patches of ideal patterns separated by domain walls.

A pivotal result of our qualitative theoretical study is the derivation of the system of globally coupled Ginzburg–Landau equations governing the dynamics of slowly varying spatiotemporal envelopes of ideal roll patterns in anisotropic systems near the experimentally observed codimension-two point. We have identified primary solution branches, studied their stability, and identified regions in parameter space giving rise to superpositions of these solutions (mixed mode solutions) in the context of an idealized normal form description restricted to spatially uniform envelopes of ideal patterns. The main features of the resulting bifurcation diagrams are that there is either a continuous transition between the two primary branches via a stable mixed mode branch, or a region with bistability and an unstable mixed mode branch leading to a hysteretic transition. Our experiments do not yet provide evidence which of the two scenarios is present in the physical system. Further experiments in which the voltage is carefully increased and decreased for _{0} _{0}_{c}

The next step in the theoretical analysis of the mode interaction will be a numerical study of the patterns predicted by the globally coupled Ginzburg–Landau equations. Of special interest here is the region in which the normal traveling waves are created in the primary instability. We expect that the two normal form scenarios described above will lead to rather different spatiotemporal patterns, which will provide further criteria allowing to distinguish between them in experiments. Ultimately, the connection between the experiment and the theoretical model has to be established by computing the coefficients of the Ginzburg–Landau equations from the equations of the weak electrolyte model for the material parameters of the Phase V sample used in the experiments. Such calculations have been performed in [

This research has been partially supported by the National Science Foundation under the Grant DMS-0407418 & DMS-0407201.

_{0} = 350 Hz corresponding to _{0}_{H}_{H}

(_{c}_{H}_{0}_{q}_{H}

Snapshots of patterns in cell A at 35 °C for (_{0} =90 Hz (_{0}_{q}_{0} =95 Hz (_{0}_{q}

(

Variation of (

Variation of _{⊥} (up triangles) and _{⊥} (circles) with _{0}_{q}_{0}_{q}

(_{s}_{or}_{r}_{r}_{1}_{r}_{or}