Feedback control of flow alignment in sheared liquid crystals

Based on a continuum theory, we investigate the manipulation of the nonequilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary “flow alignment,” under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the $i$th component $q_i\left(t\right)$ of the order parameter tensor is supplemented by a control term involving the difference $q_i\left(t\right)-q_i(t-\tau )$. In this diagonal scheme, $\tau$ is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength $K$ and $\tau$; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of $K$ are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic.

Figure 1

Dynamical state diagram in the plane spanned by the tumbling parameter ${\lambda }_{\mathrm{K}}$ and the (dimensionless) shear rate $\dot{\gamma }$ for a nematic system ($\theta =0$). The lines were obtained via a codimension-2 bifurcation analysis as described in Ref. [27]. The nearly vertical line represents a supercritical Hopf bifurcation line (H). On the right side of the H line we find flow alignment (A), corresponding to a stable fixed point. This is indicated by the notation (1s,0u), where 1s and 0u mean one stable (s) and no unstable (u) fixed point, respectively. Upon crossing the H line towards lower values of ${\lambda }_{\mathrm{K}}$, oscillatory in-plane states [wagging (W) and tumbling (T)] become stable. As indicated by the notation (0s,3u) [and (0s,1u)] there is no stable fixed point in this area, but three (one) unstable ones. The cross marks the parameter set ${\mathit{\beta }}_{\mathrm{I}}=({\lambda }_{\mathrm{K}}=1.0,\dot{\gamma }=5.0,\theta =0)$ where we apply feedback control. The shaded areas appearing at low $\dot{\gamma }$ correspond to oscillatory states with out-of-plane symmetry. The lines labeled LP1 and LP2 are limit point lines (saddle-node bifurcation lines).

Figure 2

Dynamical state diagram at $\theta =1.20$, where the unsheared system is isotropic. The lines are obtained via a codimension-2 bifurcation analysis as described in Ref. [27]. With the bifurcation analysis we also detect special points like the cusp point (CP), Bogdanov-Takens point (BT), and generalized Hopf point (GH). The solid line (shaped like an oval) represents a supercritical H bifurcation line, within which the system is in an oscillatory (W) state. The cross marks the parameter set ${\mathit{\beta }}_{\mathrm{II}}=({\lambda }_K=0.6,\dot{\gamma }=0.8,\theta =1.20)$ selected as a second candidate for feedback control. Outside this W regime (characterized by one unstable fixed point), the system is in a stationary, flow-aligned state (A). The dashed lines appearing in the bottom right corner of the diagram are limit point lines (LP) (saddle-node bifurcation lines). Between the LP curves one finds areas of bistability between paranematic and nematic states (for details on that, see Ref. [27]). The notation regarding the fixed points is as in Fig. 1.

Figure 3

Nullclines corresponding to the dynamical variables $a_0$, $a_1$, $a_2$. The parameters are fixed to the set ${\mathit{\beta }}_{\mathrm{I}}$ located within the in-plane W regime (see Fig. 1). The striped, unichrome, and checkerboard patterned surfaces are obtained from setting $\dot{a_0}=0$, $\dot{a_1}=0$, and $\dot{a_2}=0$, respectively (components $a_3$ and $a_4$ vanish at ${\mathit{\beta }}_{\mathrm{I}}$). The small spheres indicate the intersections of all nullclines and thus depict the (unstable) fixed points of the system, ${\mathit{q}}_1^{☆}=(1.30034,0.139206,0.318084,0,0)$, ${\mathit{q}}_2^{☆}=(-0.563386,0.807425,-0.217522,0,0)$, and ${\mathit{q}}_3^{☆}=(0.98609,0.232099,0.38358,0,0)$. The white cycle corresponds to the stable limit cycle emerging around ${\mathit{q}}_2^{☆}$.

Figure 4

Phase portraits of the dynamical variables $a_i$ ($i=0,1,2$). Parameters are fixed to the set ${\mathit{\beta }}_{\mathrm{I}}$ located within the in-plane W regime (see Fig. 1). The central small red dot marks the (unstable) fixed point. (a) Uncontrolled system ($K=0$) with the initial condition $a_0^{\mathrm{init}}=-0.57$, $a_1^{\mathrm{init}}=0.5$, $a_2^{\mathrm{init}}=-0.22$. The initial condition is indicated with a black diamond. (b)–(f) Systems under time-delayed feedback control [see Eq. (3.2)] with control strength (b) $K=0.1$, (c) $0.2$, (d) $0.3$, (e) $0.4$, and (f) $0.5$, respectively, and delay time $\tau =0.5$. In all cases in (b)–(f), the control starts at $t=0$, assuming that $a_i\left(t\right)=a_i^{\mathrm{init}}$ in the interval $[-\tau ,0]$. In (a)–(d) the blue (dark gray) trajectories approach stable limit cycles, which are plotted as thick red (medium gray) cycles. In (b)–(f) the limit cycle of the uncontrolled system in (a) is replotted for reference as a green (light gray) cycle. For (e) and (f) the trajectories end at the fixed point.

Figure 5

Largest real part of the complex eigenvalues $\mu$ vs $\tau$ for different values of $K$. The dot-dashed vertical lines correspond to multiples of $T_0=2\pi /\left|\mathrm{Im}\left(\nu \right)\right|$ ($T_0\approx 1.36$), where $\nu$ is the eigenvalue of the uncontrolled system at the fixed point ${\mathit{q}}_2^{☆}$ (see Fig. 3). The solid vertical lines are shifted relative to the dashed lines by $T_0/2$. The dashed lines in the lower part indicate some lower branches of eigenvalues for $K=0.30$. The other parameters are fixed to the set ${\mathit{\beta }}_{\mathrm{I}}$ located within the in-plane W regime (see Fig. 1).

Figure 6

Largest real part of the complex eigenvalues $\mu$ in the $K$-$\tau$ plane (only negative values are shown). The black contours bound regions where the largest real part is negative. The shear parameters are fixed to the set ${\mathit{\beta }}_{\mathrm{I}}$ located within the in-plane W regime (see Fig. 1).

Figure 7

Same as Fig. 3, but for the parameter set ${\mathit{\beta }}_{\mathrm{II}}$ located within the island of W dynamics, which appears on shearing from the isotropic phase (see Fig. 2). The small sphere indicates the unstable fixed point, ${{\mathit{q}}^{*}}_1=[-0.283501,0.49587,0.108511,0,0]$. The white cycle corresponds to the stable limit cycle emerging around ${\mathit{q}}_1^{☆}$.

Figure 8

Phase portraits of the dynamical variables $a_i$ ($i=0,1,2$). The central small dot marks the (unstable) fixed point. (a) Uncontrolled system ($K=0$) with initial condition $a_0^{\mathrm{init}}=-0.20$, $a_1^{\mathrm{init}}=0.4$, $a_2^{\mathrm{init}}=0.1$. (b)–(f) Systems under time-delayed feedback control [see Eq. (3.2)] with the control strength (b) $K=0.03$, (c) $0.05$, (d) $0.1$, (e) $0.15$, and (f) $0.2$, respectively, and delay time $\tau =5.0$. In all cases in (b)–(f), the control starts at $t=0$, assuming that $a_i\left(t\right)=a_i^{\mathrm{init}}$ in the interval $[-\tau ,0]$. In (a) the blue (dark gray) trajectory approaches a stable limit cycle, which is plotted as a thick red (medium gray) cycle. In (b)–(f) the limit cycle of the uncontrolled system in (a) is replotted for reference as a green (light gray) cycle. For (b)–(f) all trajectories end in the fixed point. The shear parameters are fixed to the set ${\mathit{\beta }}_{\mathrm{II}}$; see Fig. 2.

Figure 9

Largest real part of the complex eigenvalues $\mu$ vs $\tau$ for different values of $K$ (the shear parameters are fixed to the set ${\mathit{\beta }}_{\mathrm{II}}$; see Fig. 2). The dash-dotted vertical lines correspond to multiples of $T_0=2\pi /\left|\mathrm{Im}\left(\nu \right)\right|$ ($T_0\approx 9.68$), where $\nu$ is the eigenvalue of the uncontrolled system at the fixed point ${\mathit{q}}_1^{☆}$ (see Fig. 7). The solid vertical lines are shifted relative to the dash-dotted lines by $T_0/2$. The dashed lines in the lower part indicate some lower branches of eigenvalues for $K=0.03$.

Figure 10

Largest real part of the complex eigenvalues $\mu$ in the $K$-$\tau$ plane. The black contours bound regions where the largest real part is negative; these regions are colored according to the color map. The shear parameters are fixed to the set ${\mathit{\beta }}_{\mathrm{II}}$; see Fig. 2.