Energy surface and minimum energy paths for Fréedericksz transitions in bistable cholesteric liquid crystals

The multidimensional energy surface of a cholesteric liquid crystal in a planar cell is investigated as a function of spherical coordinates determining the director orientation. Minima on the energy surface correspond to the stable states with particular director distribution. External electric and magnetic fields deform the energy surface and positions of minima. It can lead to the transitions between states, known as the Fréedericksz effect. Transitions can be continuous or discontinuous depending on parameters of the liquid crystal which determine an energy surface. In a case of discontinuous transition when a barrier between stable states is comparable with the thermal energy, the activation transitions may occur, and it leads to the modification of characteristics of the Fréedericksz effect with temperature without explicit temperature dependencies of liquid crystal parameters. A minimum energy path between stable states on the energy surface for the Fréedericksz transition is found using the geodesic nudged elastic band method. Knowledge of this path, which has maximal statistical weight among all other paths, gives the information about a barrier between stable states and configuration of director orientation during the transition. It also allows one to estimate the stability of states with respect to the thermal fluctuations and their lifetime when the system is close to the Fréedericksz transition.

Figure 1

Stable states of a ChLC confined in a planar cell. (a) Planar (P) state, when the director is perpendicular to the helix axis $\left(\theta =\pi /2,\varphi =q_{0}z\right)$. (b) Distorted (D) state, when the director has nonzero projections on the helix axis $[\theta =\pi /2+\delta \theta (z),\varphi =q_{0}z+\delta \varphi \left(z\right)]$. $\delta \theta \left(z\right),\delta \varphi \left(z\right)$ are found from a direct minimization of the total energy of the ChLC [see Eq. (1)]. External electric (or magnetic) field is along the $z$ axis.

Figure 2

Energy of the ChLC as a function of applied voltage (a) and magnetic field (b). Curves P and D correspond to planar and distorted states, respectively. Metastable states are shown with dashed curves.

Figure 3

Energy surface of the ChLC in a magnetic field as a function of the first nonzero Fourier components, $c_{\theta }$ and $c_{\varphi }$, of the spherical coordinates $\theta \left(z\right)$ and $\varphi \left(z\right)$ defining orientation of the director. (a) $H<H^{**}$. (b) $H^{**}<H<H^{*}$. The MEPs are shown with a white curve, and SPs are indicated with crosses.

Figure 4

Energy per unit area along the MEP at $U=U_c=986.84$ mV. The filled circles correspond to the images of the system used in the GNEB calculation. The reaction coordinate is defined as the displacement along the path normalized by its total length.

Figure 5

Profiles of the polar angle (a) and deviation of the azimuthal angle from a straight line, $\varphi \left(z\right)=q_{0}z$, (b) for the images along the MEP. Labeling of the curves corresponds to that of the points in Fig. 4. Inset in (b): $\mathrm{\Delta }\varphi \left(z\right)$ for the opposite twist of the ChLC.

Figure 6

(a) MEPs for the transition between P and D states at different applied voltages. The reaction coordinate is defined as the displacement along the path normalized by its total length. (b) Energy barrier for transition from the P to D state (solid line) and for the inverse transition (dashed line) as a function of applied voltage.

Figure 7

(a) The MEPs for transition from the P to the D state for different anchoring coefficients $W_{\varphi }^u$ at the upper boundary. The reaction coordinate is defined as the displacement along the path normalized by its total length. (b) Energy barrier for transition from the P to D state (solid line) and for the inverse transition (dashed line) as a function of $W_{\varphi }^u$. Voltage is taken to be $U=987\mathrm{mV}$.