Stability of the Cholesteric Phase Exposed to a Magnetic Field

Abstract

It is shown that the cholesteric phase is stabilized by the application of magnetic field. Without the field, the fluctuation of molecular orientations is developed to lead to the instability. The breaking of rotational symmetry about the twist axis by application of field suppresses the fluctuation to retain the stability. It has been shown by Lubensky1l that the fluctuation of molecular orientation within a plane perpendicular to the twist axis is too large to keep the cholesteric phase to be stable. Furthermore, he has suggested that a finite system is stabilized owing to the boundary effect. In this short note, we show that the stability of the cholesteric phase is retained by the application of magnetic field even with no boundary effect taken into account. In the absence of magnetic field, the free energy of the cholesteric phase is invariant with respect to a rotation around the twist axis. This invariance or the rotational symmetry is violated by the magnetic field applied in perpendicular to that axis, and consequently there ap­ pears a gap in the energy spectrum of the orientational fluctuation. The gap varies continuously along the twist axis, being possible to be not only positive but also negative. The fluctuation of long wave length of the system size is relevant to determine the stability of the cholesteric phase. The gap in the energy spectrum is locally averaged over a period of the pitch in such a fluctuation and the average is shown to be definitely positive. Thus, the cholesteric phase exposed to an exter­ nal magnetic field is concluded to be stable. The free energy of the cholesteric phase can be expressed in terms of the director field n (r) as2 l