Instability pattern forming

O. Carrillo, M. A. Santos, J. GarciarOjalvo, and J. M. Sancho. Spatial coherence resonance near pattern-forming instabilities. Europhys. Lett, 65 452, 2004. 

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. 

W. Pesch and L. Kramer, General Mathematical Description of Pattern-Forming Instabilities. In Pattern Formation in Liquid Crystals, editors A. Buka and L. Kramer, pages 69-90, Springer, New York, 1996. 

The main purpose of this paper is to present a simple theoretical model to describe this interesting phenomenon which provides a further example of the synergy between an equilibrium phase transition and a pattern forming instability. It is based on a Landau type equation for the polymer volume fraction coupled to a simple two variables reactive system that, on its own, can undergo a Hopf bifurcation giving rise to chemical oscillations of the limit cycle type. These oscillating systems have now been extensively studied both from the theoretical and experimental points of view (7). 

Besides the elastic and the electric torques the so-called flexoelectric (or flexo) torques on the director play an important role as well. Their effect on pattern-forming instabilities in nematics is the main issue of this chapter. Flexotorques originate from the fact that typically (in some loose analogy to piezoelectricity) any director distortion is accompanied by an electric flexopolarization Pa (characterized by the two ffexocoefScients ei, 63). From a microscopic point of view, finite ei and 03 naturally arise when the nematic molecules have a permanent dipole moment. But also for molecules with a quadrupolar moment, finite ei and 63 are possible (see also Chapter 1 in this book ). Flexopolarization has to be incorporated into the free energy P n) for finite E. It is not surprising that this leads to quantitative modifications of phenomena, which exist also for ci = 63 = 0. Though, for example, the Freedericksz threshold field Ep is not modified, the presence of flexoelectricity leads to considerable modifications of the Freedericksz distorted state for E > Ep- ... 

Electroconvection in nematics is certainly a prominent paradigm for nonequilibrium pattern-forming instabilities in anisotropic systems. As mentioned in the introduction, the viscous torques induced by a flow field are decisive. The flow field is caused by an induced charge density p i when the director varies in space. The electric properties of nematics with their quite low electric conductivity 10 (fl m) ] are well described within the electric quasi-static approximation, i.e. by charge conservation and Pois-... 

E. Bodenschatz, W. Zimmermann and L. Kramer, On electrically driven pattern-forming instabilities in planar nematics, J. Phys. France 49(11), 1875-1899, (1988). doi 10.1051/jphys 0198800490110187500... 

W. Pesch, L. Kramer, General mathematical description of pattern forming instabilities, in A. Buka, L. Kramer (eds.) Pattern formation in liquid crystals. Springer, New York, (1995)... 

S. R. Lee and J. S. Kim, On the sublimit solution branches of the stripe patterns formed in counterflow diffusion flames by diffusional-thermal instability. Combust. Theory Model. 6(2) 263-278,2002. 

In the second group we find pattern forming phenomena based on new instability mechanisms arising from the specific features of liquid crystals, which have no counterpart in isotropic fluids or at least are difficult to assess. Some examples are shear (linear, elliptic, oscillatory, etc.) induced instabilities, transient patterns in electrically or magnetically driven Freedericksz transitions, structures formed in inhomogeneous and/or rotating electric or magnetic fields, electroconvection (EC), etc. [5-7]. 

For the sake of completeness we briefly recall some basic features about the pattern forming Turing instability. The standard reaction-diffusion equations can be written in the form... 

The nuclei of some elements are stable, but others decay the moment they are formed. Is there a pattern to the stabilities and instabilities of nuclei The existence of a pattern would allow us to make predictions about the modes of nuclear decay. One clue is that elements with even atomic numbers are consistently more abundant than neighboring elements with odd atomic numbers. We can see this difference in Fig. 17.11, which is a plot of the cosmic abundance of the elements against atomic number. The same pattern occurs on Earth. Of the eight elements present as 1% or more of the mass of the Earth, only one, aluminum, has an odd atomic number. 

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