Robustness of the periodic and chaotic orientational behavior of tumbling nematic liquid crystals

The dynamical behavior of molecular alignment strongly affects physical properties of nematic liquid crystals. A theoretical description can be made by a nonlinear relaxation equation of the order parameter and leads to the prediction that rather complex even chaotic orientational behavior occur. Here the influence of fluctuating shear rates on the orientational dynamics especially on chaotic solutions is discussed. With the help of phase portraits and time evolution diagrams, we investigated the influence of different fluctuation strengths on the flow aligned, isotropic, and periodic solutions. To explore the effect of fluctuations on the chaotic behavior, we calculated the largest Lyapunov exponent for different fluctuation strengths. We found in all cases that small fluctuations of the shear rate do not affect the basic features of the dynamics of tumbling nematics. Furthermore, we present an amended potential modeling the isotropic to nematic transition and discuss the equivalence and difference to the commonly used Landau–de Gennes potential. In contrast to the Landau–de Gennes potential, our potential has the advantage to restrict the order parameter to physically admissible values. In the case of extensional flow, we show that the amended potential leads for increasing extensional rate to a better agreement with experimental results.

Figure 1

The Landau–de Gennes potential (gray line) and the amended potential (dark line) as a function of the scalar order parameter $a$ for $\vartheta =0$ (left) and $\vartheta =1$ (right).

Figure 2

The rheological phase plot is shown for flow alignment (A), in-plane (T/W), and out-plane regimes (symmetry breaking states). In the upper panel, the amended potential and in the lower panel the Landau–de Gennes potential was used. The reduced temperature is $\vartheta =0$ and the parameter $\kappa =0$.

Figure 3

The average of the scalar order parameter $a=\sqrt{\mathbf{a}:\mathbf{a}}$ calculated numerically is plotted vs the extensional rate $\dot{ϵ}∕6$. The points are the results from the numerical solution of the Fokker-Planck equation obtained in [26]. The orientation increases without limits for the Landau–de Gennes potential (black line), whereas it saturates for the amended potential (gray line). The model parameter are ${\lambda }_{\mathrm{K}}=1$, $\kappa =0$, and $\vartheta =0$.

Figure 4

The average of the scalar order parameter $a=\sqrt{\mathbf{a}:\mathbf{a}}$ as a function of the extensional rate is displayed for different parameter values $\kappa$ ($\kappa =0.4$ dark curve, $\kappa =0.8$ dashed curve, $\kappa =1$ gray curve). The model parameters are ${\lambda }_{\mathrm{K}}=4∕3$ and $\vartheta =0$.

Figure 5

Pseudorandom fluctuation given by $f\left(t\right)$ as a nonlinear function of the time $t$, Eq. (47).

Figure 6

The distribution function of dimensionless pseudorandom fluctuations $x$ is displayed [left for numerical calculated random numbers and right for random numbers calculated via the nonlinear function $f\left(t\right)$].

Figure 7

The scalar order parameter $a$ in the isotropic regime as a function of the time is given for several values of the parameter $\xi$ (pentagonal for $\xi =0$, triangle for $\xi =5$, and cross for $\xi =10$). The reduced temperature is $\vartheta =1$; the model parameters are ${\lambda }_{\mathrm{K}}=1.0$, ${\dot{\gamma }}_0$ and $\kappa =0$.

Figure 8

The time evolution of the scalar order parameter $a$ in the flow alignment regime with fluctuations of the shear rate $(\xi =2.5)$ around ${\gamma }_0=2.0$ calculated numerically and analytically (dashed line) compared to the original solution (gray line, $\xi =0$) (left). The time evolution of the tensor component $a_1$ in the flow alignment regime for strong fluctuations $(\xi =5.0)$ of the shear rate around $\dot{{\gamma }_0}=2.0$ and in the absence of fluctuations (gray line) is compared (right). The reduced temperature is $\vartheta =0$; the model parameters are ${\lambda }_{\mathrm{K}}=1.5$ and $\kappa =0$.

Figure 9

The orbits $a_4$ vs $a_3$, $a_2$ vs $a_1$, and the time evolution of $a_3$ is plotted for several coupling constants: $c=0$ (upper left), $\xi =50$ (upper right) and $\xi =170$ (lower left). In the lower right, the graphs for $\xi =0$ (dashed line), $\xi =50$, and $\xi =170$ (gray line) are displayed. The temperature is $\vartheta =0$; the model parameters are ${\lambda }_{\mathrm{K}}=1.0$, $\dot{{\gamma }_0}=1.0$ and $\kappa =0$.

Figure 10

The orbit $a_1$ vs $a_2$ at fluctuation strength $\xi =10$ of the shear rate around $\gamma =1.0$ are displayed. The fluctuations are modelled numerically (left) and analytically (right). The reduced temperature is $\vartheta =0$; the model parameters are ${\lambda }_{\mathrm{K}}=1.0$ and $\kappa =0$.

Figure 11

The orbits $a_1$ vs $a_2$ in the tumbling regime for $\xi =0$ upper left, $\xi =1$ upper right, and $\xi =5$ lower left. In the lower right the time evolution of $a_1$ for $c=5$ is given. The reduced temperature is $\vartheta =0$; the model parameters are ${\lambda }_K=0.8$, $\dot{\gamma }=6$, and $\kappa =0$.

Figure 12

The time evolution of the scalar order parameter $a$ with $(\xi =2.5)$ and without (gray line) fluctuations. The reduced temperature is $\vartheta =0$; the model parameters are ${\lambda }_{\mathrm{K}}=1.0$, $\dot{{\gamma }_0}=1.0$, and $\kappa =0$.

Figure 13

The largest Lyapunov exponent is plotted against the amplitude of fluctuations $\xi$. Fluctuations are modeled as Wiener process. The reduced temperature is $\vartheta =0$; the model parameters are ${\lambda }_{\mathrm{K}}=1.17$, ${\dot{\gamma }}_0=3.72$, and $\kappa =0$.

Figure 14

The largest Lyapunov exponent is plotted against the amplitude of fluctuations $\xi$, where fluctuations are modeled by Eq. (47). The same parameters as in Fig. 12 are chosen.

Figure 15

The orbit $a_1$ vs $a_2$ (left) and $a_3$ vs $a_4$ (right) in the chaotic regime at mean shear rate ${\gamma }_0=3.7$ for several coupling constants $\xi$ (upper graph $\xi =0.05$ and lower graph $\xi =8$). The reduced temperature is $\vartheta =0$; the model parameters are ${\lambda }_{\mathrm{K}}=1.15$ and $\kappa =0$.