ELECTROHYDRODYNAMIC INSTABILITY IN PLANAR, POSITIVE DIELECTRIC ANISOTROPY NEMATIC LAYERS AT D.C. EXCITATION

Abstract

A generalisation of de Gennes model for the electrohydrodynamic instability at strong unipolar injection is developed. Including in the torque balance equation the elastic torque gives the possibility for description of positive materials as well. Two domains of dielectric anisotropy values (small and big) are found with qualitatively different behaviour of the instability. The case of no injection is considered as well. In conclusion some other mechanisms, active at d.c. excitation are briefly discussed, especially the flexo-electric effect-gradient and linear one. 1 . Theory. The considerations given below originate from a model developed by de Gennes in 1970 [I]. Let a nematic, placed in a planar capacitor with interelectrode distance d and oriented parallel to the electrodes, is subjected to a strong unipolar injection by a d.c. voltage V, applied to the electrodes. The condition for strong injection means that the density of injected charges go near to the injecting electrode (the cathode usually) is big enough to compensate a great part of the applied electric field in that region. Numerically it means that The development of an instability will be studied on the basis of the charge balance equation. The time evolution of a charge density fluctuation is described by where an assumption is made for the convective term at strong injection that In the second term 7 is the dielectric relaxation time : o is the mean value of the conductivity and the coefficient 2 arises because the conductivity itself is proportional to 69 [I]. The third term is due to the Carr-Helfrich effect. The transverse current is determined by the angle of deviation cp of the director from the Ox axis. For small angles this connection is In order to express the fluid velocity v, via 69 the Stokes equation is used (with neglecting of the inertial effects) : q : mean viscosity of the nematic. In the case of spatial variations of v, in the following form (determined by the presence of electrodes) : 2 nx v, = vt cos ( , ) cos (7) , (6) Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979359 ELECTROHYDRODYNAMIC INSTABILITY IN POSITIVE NEMATICS C3-311 it follows from (5) that In the original version of the model the deviation angle cp was calculated from a torque balance equation containing only dielectric (re) and hydrodynamic (T,) torque with neglecting of the elastic one. The complete equation should read Here rk,,, the y-component of the elastic torque, in isotropic elasticity approximation (K,, = K,, = K) is The last result can be obtained if the same form as for v, is adopted for the spatial variation of 9 : (P = cp, cOS (T) cos (T) . (9) With de Gennes' results for both other torques the torque balance equation reads A differentiation of this equation with respect to x together with de Gennes' estimation : and the result from (7)