Wavenumber, critical

Abstract A systematic overview of various electric-field induced pattern forming instabilities in nematic liquid crystals is given. Particular emphasis is laid on the characterization of the threshold voltage and the critical wavenumber of the resulting patterns. The standard hydrodynamic description of nematics predicts the occurrence of striped patterns (rolls) in five different wavenumber ranges, which depend on the anisotropies of the dielectric permittivity and of the electrical conductivity as well as on the initial director orientation (planar or homeotropic). Experiments have revealed two additional pattern types which are not captured by the standard model of electroconvection and which still need a theoretical explanation. 

Figure 2. Threshold voltage Uth/Uo and the critical wavenumber qc versus the dimensionless dielectric anisotropy eajev calculated from Eqs. (7) and (8). a b Planar alignment with a a > 0, c d homeotropic ahgnment with a a < 0. Dashed lines correspond to the Freedericksz transition, solid lines to the direct EC transition.

In Fig. 2, the results for the critical voltage Uth (left panels, in units of Uq = /n Ki/ eoe ), Uq = 1.19V for MBBA parameters) and the corresponding critical wavenumber Qc (right panels) are summarized as functions of e /e L. The data are barely distinguishable from the results of a rigorous linear stability analysis based on the full standard model [21]. 

Figure 7. Threshold voltages Uth/Co and the critical wavenumber qc versus the relative dielectric anisotropy eajei. calculated from Eq. 8. Homeotropic alignment with <7a > 0. The upper (a b ) and lower (c d ) plots differ only in the axis scales. Dashed lines correspond to the Freedericksz transition, solid lines correspond to the direct transition to an ("a-induced") EC patterned state, dotted lines represent a secondary transition to EC.

Typical dispersion curves defined by (39) are qualitatively the same as shown in Fig.2b. Note that the critical wavenumber at the threshold does not depend on the wetting potential and is determined only by the surface stiffness and the energy of edges and corners. For the parameter values typical of semiconductors like Si or Ge, with the surface energy 7 2.0 Jm , surface stiffness a 0.2 Jm , the lattice spacing ao 0.5 nm and the regularization parameter u Oq 5.0 X 10 J, the wavelength of the structure at the onset of instability is 14.0 nm. 

Theorem 10.4 The critical wavenumber kj of the Turing instability near a doublezero point is given by... 

The Turing condition for two-variable DIRWs, (10.85), has the same form as the Turing condition for two-variable reaction-diffusion systems, (10.31). Consequently, the uniform steady state (10.70) of a DIRW undergoes a Turing bifurcation with critical wavenumber... 

This curve has a single minimum, (A x, bj), which corresponds to the Turing instability of the uniform steady state. The Turing threshold bj and the critical wavenumber kj depend on q and read in parametric form... 

Under these conditions there is no oscillation and the disturbance dther grows or decays exponentially, depending on the sign of. The critical wavenumber a, is still given by Equation 5.52, but there is no finite a. The smaller the value of a(consistrait with ip/Vga l 1), the greater the rate of growth in the unstable region. 

For situations where the film is unstable. Equation 5.116 can be solved for the critical wavenumber for which p vanishes. 

Note that the nth mode has a critical wavenumber of k /n, and a maximum growth rate of (a2/4y) - independent of n - at k /(n V2). This implies that unstable modes will be stabilized by energy transfer to higher harmonics. 

As (57) shows the longitudinal oscillations u carry surface active agent out of the points where c is minimal and bring it to the points where c is maximal. Some sort of resonance takes place. The unbounded solutions appear at the critical wavenumber defined by (54). When Ma < 0 and Ma is small the critical wavenumber of the explosive models outside of the hydrodynamic mode instability interval exposed (a > a ), with Ma growing ak moves to... 

Clearly, BBc instability. For Type 2 instability, the critical B value (denoted as and critical wavenumber q are obtained from the conditions fi(qc) = 0 and dp qc)/dqc = 0. We have... 

Let us assume that an > 0,21- Furthermore, if an >0 and still subject to a n + a22 < 0, then it is clear that Re A1 becomes positive at sufficiently large k and that the system becomes unstable against short wavelength perturbations. The critical wavenumber kc (i.e. such that Re Ai(fcc) = 0) is... 

If R P) is negative then that particular modulation will not grow and from equation (23) the critical wavenumber is given by... 

The roots of this characteristic equation give two critical wavenumbers ... 

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