Magnetic-field-induced stepwise director reorientation and untwisting of a planar cholesteric structure with finite anchoring energy

Within the continuum approach we study the equilibrium configurations of a cholesteric liquid crystal confined between two parallel plates, when a magnetic field is applied perpendicularly to the plates. We analyze the role of soft anchoring boundary conditions on magnetic-field-induced cholesteric-nematic transitions in a finite thickness cholesteric cell, treated to induce soft planar alignment. We study the stepwise behavior of cholesteric pitch as a function of the anchoring energy, the thickness of a layer, and the field strength. We analyze some kinds of soft anchoring potentials, including the case of degeneration of the easy axes. We show that the variation of the thickness or intrinsic pitch induces the the stepwise behavior of a pitch of the planar cholesteric structure, and the stepwise variations of the average tensor of diamagnetic susceptibility. The values of these jumps are determined by the anchoring energy. We find the values of critical parameters for the transitions between planar and confocal cholesteric states, and homeotropic nematic state. We show that the variation of the anchoring energy leads to change of the phase transition character; the conditions for hysteresis behavior are obtained. We show that for rather soft anchoring the confocal state is metastable, and the increase of a magnetic field leads to the direct transition between the planar cholesteric and homeotropic nematic phases. We also give a detailed derivation of the threshold and saturation properties of planar cholesteric to homeotropic nematic transition.

Figure 1

Dependence of ${\mathcal{F}}_0$ on the thickness of the layer $D$ for $k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: (a) $w=0.3$, (b) $w=1$, (c) $w=3$, and (d) $w=5$.

Figure 2

Dependence of the pitch of spiral $p$ on the thickness of the layer $D$ in the planar state for $k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: (a) $w=0.3$, (b) $w=1$, (c) $w=3$, and (d) $w=5$.

Figure 3

Dependence of ${\phi }_S$ on the layer thickness $D$ in the planar state for $k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: (a) $w=0.3$, (b) $w=1$, (c) $w=3$, and (d) $w=5$.

Figure 4

Dependence of $H_F$ and $H_S$ (the curve with the index “S”) on the thickness of the layer $D$ for $w=5, k_2=0.6, k_3=1.5$, and $\beta =\pi /4$.

Figure 5

Dependence of $H_F$ and $H_S$ (the curve with the index “S”) on the thickness of the layer $D$ for $w=0.3, k_2=0.6, k_3=1.5$, and $\beta =\pi /4$.

Figure 6

Dependence of the angle ${\theta }_m$ on the magnetic field strength $H$ for $w=5, D=8, k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: curve 0, $n=0$, angle ${\phi }_m=0$; curve 1, $n=1$, angle ${\phi }_m=\pi /2$; curve 2, $n=2$, angle ${\phi }_m=\pi$.

Figure 7

Dependence of ${\mathcal{F}}_{\theta }$ on the magnetic field strength $H$ for $w=5, D=8, k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: curve 0, $n=0$, angle ${\phi }_m=0$; curve 1, $n=1$, angle ${\phi }_m=\pi /2$; curve 2, $n=2$, angle ${\phi }_m=\pi$.

Figure 8

Dependence of the angle ${\theta }_m$ on the magnetic field strength $H$ for $w=5, D=5, k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: curve 0, $n=0$, angle ${\phi }_m=0$; curve 1, $n=1$, angle ${\phi }_m=\pi /2$.

Figure 9

Dependence of ${\mathcal{F}}_{\theta }$ on magnetic field strength $H$ for $w=5, D=5, k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: curve 0, $n=0$, angle ${\phi }_m=0$; curve 1, $n=1$, angle ${\phi }_m=\pi /2$.

Figure 10

Dependence of $H_t^{*}$ on the layer thickness $D$ for $k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: curve 1, $w=0.3$; curve 2, $w=1$.

Figure 11

Dependence of $H_F$ (thick lines) and $H_S$ (thin lines) on the inverse anchoring energy $1/w$ for $k_2=0.6, k_3=1.5$, and $\beta =\pi /4$: curve 1, $D=8$; curve 2, $D=5$; curve 3, $D=2$; curve 4, $D=1$.

Figure 12

Dependence of the field $H_F$ on the thickness $D$ for $k_2=0.6, k_3=1.5, \beta =\pi /4$, and $w_a=5$: dashed lines, $w_p=0$; solid lines, $w_p=4.5$.

Figure 13

Dependence of the field $H_S$ on the thickness $D$ for $k_2=0.6, k_3=1.5, \beta =\pi /4$, and $w_a=5$; the numbers in the curves correspond to $w_p$.

Figure 14

Dependence of the Fréedericksz field $H_F$ and saturation field $H_S$ (dashed line) on the thickness $D$ of the layer for $w=5, k_2=0.8, k_3=2.0$, and $\beta =\pi /4$. The integer indices are the values of $n$.