Patterns and Defects in Liquid Crystals

Bodenschatz, Eberhard; Kaiser, Matthias J.; Kramer, Lorenz; Pesch, Werner; Weber, Andreas; Zimmermann, Walter

doi:10.1007/978-1-4684-7479-4_20

Cited by 12 publications

(8 citation statements)

References 21 publications

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“…Presumably the system goes directly into a complex spatio-temporal chaotic state similar to the case of planarly aligned nematics with fi.ee boundary conditions [7,8]. This state of weak turbulence should be suppressible by a planar magnetic field.…”

mentioning

confidence: 99%

The electrohydrodynamic instability in homeotropic nematic layers

Hertrich¹,

Decker²,

Pesch³

et al. 1992

J. Phys. II France

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We calculate the threshold for electrohydrodynamic convection in homeotropically oriented nematic layers with negative dielectric anistropy that sets in after the bend-Fréedericksz transition to a quasi-planar alignment has taken place. Oblique rolls are found in a larger range than in the case of planar anchoring. The most interesting prediction is that in the weakly nonlinear analysis all roll solution are in fact unstable so that a direct transition to spatio-temporal chaos becomes possible. In the oblique-roll case this may be accompanied by a permanent local rotation of the director bend axis. Application of a planar magnetic field should stabilize the rolls

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mentioning

confidence: 99%

The electrohydrodynamic instability in homeotropic nematic layers

Hertrich¹,

Decker²,

Pesch³

et al. 1992

J. Phys. II France

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“…In two dimensions phase conservation is less common than in one dimension due to the possibility of focus singularities in the pattern which often arise at the boundaries [6]. Our results may therefore also shed some additional light on the behavior of the 'zig-zag' patterns studied previously [17,3].…”

Section: Introductionmentioning

confidence: 67%

“…Domain structures arise for negative values of D. For large values of Σ wave-number gradients across domain walls are sufficiently small to be described by (3), and within this framework the interaction between domain walls is purely attractive. Thus, in the presence of a subcritical ramp domain structures are expected to evolve to a strictly periodic pattern.…”

Section: Numerical Simulation Of Fourth-order Ginzburg-landau Equationmentioning

confidence: 99%

“…Thus, the domain structures discussed in sec.2 are one-dimensional analogs of zig-zag patterns. The general stability of zig-zag patterns in anisotropic systems has been studied by Pesch and Kramer [17] and by Bodenschatz et al [3] who find islands of stability for the zig-zag states. They are presumably related to the discrete set of domains with different widths discussed above.…”

Section: Application To Two Dimensional Systems: Zig-zagsmentioning

confidence: 99%

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Domain structures in fourth-order phase and Ginzburg-Landau equations

Raitt

Riecke

1995

Physica D: Nonlinear Phenomena

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In pattern-forming systems, competition between patterns with different wave numbers can lead to domain structures, which consist of regions with differing wave numbers separated by domain walls. For domain structures well above threshold we employ the appropriate phase equation and obtain detailed qualitative agreement with recent experiments. Close to threshold a fourth-order Ginzburg-Landau equation is used which describes a steady bifurcation in systems with two competing critical wave numbers. The existence and stability regime of domain structures is found to be very intricate due to interactions with other modes. In contrast to the phase equation the Ginzburg-Landau equation allows a spatially oscillatory interaction of the domain walls. Thus, close to threshold domain structures need not undergo the coarsening dynamics found in the phase equation far above threshold, and can be stable even without phase conservation. We study their regime of stability as a function of their (quantized) length. Domain structures are related to zig-zags in two-dimensional systems. The latter are therefore expected to be stable only when quenched far enough beyond the zigzag instability.

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“…The stability of these solutions as well as zigzag and undulated solutions have been investigated. 4,5 Equation (1) …”

Section: Propagating Fronts Near a Lifshitz Pointmentioning

confidence: 99%

Propagating fronts near a Lifshitz point

Zimmermann

1991

Phys. Rev. Lett.

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Dee and Saarloos 1 found in an extended Fisher-Kolmogorov (EFK) equation the transition from a uniformly translating front to a pattern-generating envelope front. A new 6 3/4 scaling of the front velocity is described and electrohydrodynamic convection (EHC) in planar nematic liquid crystals is proposed as an appropriate system for investigating some aspect of this front type.In EHC one expects for some nematic materials as a function of the frequency co of the applied voltage a continuous transition form oblique to normal-oriented convection rolls. 2 " 6 Near the transition frequency co z (Lifshitz point) the weakly nonlinear behavior is described by the amplitude equation 3 " 5 r 0 d t A=[e + ^d 2 -iZ^d x d 2 + Wtidf-tfd*-\A\ 2 ]A; (1) e = (V 2 -V 2 )/V 2 measures the distance from the normal-roll threshold V c . Numerical values for the relaxation time To and the coherence lengths §/ and Z are given elsewhere. 3 ' 5,6 W is proportional to co -co z . For periodic solutions A ^EexpliiQx+Py)], with2 and e c =0 at gc =^c =0 for normal rolls. The stability of these solutions as well as zigzag and undulated solutions have been investigated. 4,5 Equation (1) is more general then the EFK equation in Ref. 1. It is an equation for a complex field A instead of a real field and includes two space dimensions. For real ^-dependent functions Eq. (1) reduces to the EFK equation in Ref. 1, however, the scaling used there is restricted to W > 0 and 6 = 1. Using the ansatz A =exp[/(gx + 0)]Z?(y), with B a real function, then the marginal-stability analysis, described, for example, in Refs. 1 and 7, gives the velocity for fronts propagating in the y direction (parallel to the convection rolls). For U> Vl2f and U = W+QZ the uniformly translating fronts move with the velocity v\ l [36eU+U 3 -(U 2 -12e) 3/2 V nl (2) V54 TO and for U < Vl27 the envelope fronts move with velocity 2 24e + 2U 2 -FU £3 v 2 s (3) V54 V4C/ + F T o with F = (7U 2 + 24€) l/2 -3U. The wavelength of the periodic patterns behind the front for U < Vl2f is, according to the arguments used in Refs. 1 and 7, 1/2 . _ 8;r *~ 3 24e + 2U 2 -~FU \2e-U 2 -FU This wavelength X diverges for U-£3. + Vl2l.For W<0 and Q -Q c at 6> e c , however, P = 2;rA reaches the critical wave number P c of the oblique periodic solution.One could call U == -\/\2e the Lifshitz point for the transient periodic pattern the behind fronts, similar to W= § for the stationary periodic pattern. For U=0 the first nonvanishing derivative with respect to y is of fourth order in Eq. (1) and then v 2 = f x(i) 3/ V /4 £ 3 /r 0 . Very remarkable is the e 3/4 scaling, which is different to the well-known front velocity v x = 2£\y/e/Vo perpendicular to the convection rolls and to v 2 -(e-e c ) U2 for t/»0 and l/«;0 parallel to the rolls. The measurement of this new e 3/4 scaling seems more easily feasible than the transient periodicity behind the front. A possibility to detect this e law is to measure the growth velocities (v x ,v y -V2) of nuclei of convection rolls, as done in Ref. 8 apart from the Lif...

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Patterns and Defects in Liquid Crystals

Cited by 12 publications

References 21 publications

The electrohydrodynamic instability in homeotropic nematic layers

The electrohydrodynamic instability in homeotropic nematic layers

Domain structures in fourth-order phase and Ginzburg-Landau equations

Propagating fronts near a Lifshitz point

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