Oscillatory motion of sheared nanorods beyond the nematic phase

We study the role of the control parameter triggering nematic order (temperature or concentration) on the dynamical behavior of a system of nanorods under shear. Our study is based on a set of mesoscopic equations of motion for the components of the tensorial orientational order parameter. We investigate these equations via a systematic bifurcation analysis based on a numerical continuation technique, focusing on spatially homogeneous states. Exploring a wide range of parameters we find, unexpectedly, that states with oscillatory motion can exist even under conditions where the equilibrium system is isotropic. These oscillatory states are characterized by a wagging motion of the paranematic director, and they occur if the tumbling parameter is sufficiently small. We also present full nonequilibrium phase diagrams in the plane spanned by the concentration and the shear rate.

Figure 1

Dynamical state diagram in the plane spanned by the tumbling parameter ${\lambda }_{\mathrm{K}}$ and the shear rate $\dot{\gamma }$ at the reduced temperature $\theta =0$ (and $\sigma =0$). Areas are computed via direct numerical integration of the ODEs [see Eqs. (2.12) and (2.12)], whereas the curves are found from a codimension-2 bifurcation analysis. The areas are colored and labeled according to the dynamical state ($\mathrm{A}=\mathrm{alignment}$, $\mathrm{W}=\mathrm{wagging}$, $\mathrm{T}=\mathrm{tumbling}$, $\mathrm{KW}=\mathrm{kayaking}$-wagging, $\mathrm{KT}=\mathrm{kayaking}$-tumbling, and $\mathrm{C}=\mathrm{complex}$ or chaotic). The bifurcation branches are labeled as $\mathrm{LP}=$ limit point (saddle node), $\mathrm{H}=\mathrm{Hopf}$, $\mathrm{PD}=$ period doubling, $\mathrm{LPC}=$ limit point of cycles, $\mathrm{BPC}=$ branch point of cycles, and $\mathrm{NS}=\mathrm{Neimarck}$-Sacker. In addition, the diagram involves some special codimension-2 bifurcation points. These are the $\mathrm{CP}=$ cusp point, $\mathrm{CPC}=$ cusp point of cycles, $\mathrm{ZH}=$ zero Hopf, and $\mathrm{BT}=\mathrm{Bogdanov}$-Takens point.

Figure 2

Enlarged views of the parameter section of Fig. 1 in which bistability occurs. The data in (a) were obtained by sweeping the shear rate from higher to lower values, whereas in (b), the shear rate was swept from lower to higher values. In each step, the final values of the integration is used as an initial condition for the next integration step. To avoid staying on unstable solutions, the initial condition was perturbed by a small noise.

Figure 3

State diagram at the reduced temperature $\theta =1.20$, where the equilibrium state ($\dot{\gamma }=0$) is isotropic. The curves are obtained from a codimension-2 bifurcation analysis. The colors here reflect the value of the eigenvalue $\mu ={\mu }_3$, which may be considered as the degree of the uniaxial ordering (see the color bar on the right side of the figure). One observes an “island” of wagging (W) states. Region A1 corresponds to stable, shear-induced nematic order (flow alignment). In region A2, there is a coexistence of a paranematic and a nematic state. A closer view of this region (revealing even more states) is given in Fig. 5. The thin red line (the dark gray vertical line at the bottom) indicates the path of the codimension-1 bifurcation, for fixed ${\lambda }_K=0.7$, that is discussed in Fig. 4.

Figure 4

Codimension-1 bifurcation diagram with the shear rate $\dot{\gamma }$ as control parameter and the largest eigenvalue $\mu ={\mu }_3$ of the alignment tensor as the order parameter. The analysis is performed along the thin red line shown in Fig. 3 and Fig. 5 ($\theta =1.20,{\lambda }_K=0.7,\sigma =0$). The solid lines depict stable fixpoints. Dashed lines indicate unstable fixpoints. Between the homoclinic bifurcation [Hom, brown (leftmost vertical line in the center)] and the supercritical Hopf bifurcation (H), there are stable limit cycles corresponding to the wagging state (W). These limit cycles are illustrated by vertical lines between the maximal and minimal values of the limit cycles for the projection on the largest eigenvalue $\mu ={\mu }_3$.

Figure 5

Enlarged view of the parameter section of Fig. 3 at the border of the wagging island and the A2/A1 regions ($\theta =1.20$). Within the region labeled A3, stationary alignment (with low value of the order parameter) coexists with an oscillatory (W) solution.

Figure 6

Nonequilibrium phase diagram in the plane spanned by the concentration (bottom horizontal axis) or temperature (top horizontal axis) and the shear rate (vertical axis) at fixed tumbling parameter ${\lambda }_{\mathrm{K}}=1.25$. The colors represent the eigenvalue $\mu ={\mu }_3$ of the alignment tensor $\mathit{a}$. We also show the bifurcation lines obtained by a codimension-2 analysis. In the triangular area labeled A2, we find bistability between a paranematic and a nematic state with flow alignment.

Figure 7

Same as in Fig. 6 but for ${\lambda }_{\mathrm{K}}=0.74$. In the area right of the Hopf bifurcation curve (H) we observe oscillatory states. Specifically, these are wagging-tumbling (W/T) in the area between the Hopf (H) and the period doubling (PD) bifurcation curve, and kayaking-tumbling (KT) in the area right of the PD curve. In between the LPC and the PD curve, we find bistability between these states.

Figure 8

Nonequilibrium phase diagram for ${\lambda }_{\mathrm{K}}=0.6$. Compared to Fig. 7 (${\lambda }_{\mathrm{K}}=0.74$), the wagging region (W) is more extended towards the concentration (temperature) region, where the equilibrium state is isotropic [i.e., left of the red (leftmost) vertical line]. The thin white horizontal line indicates the path for the codimension-1 diagram presented in Fig. 9.

Figure 9

Codimension-1 diagram using the bifurcation parameter $c$ ($\theta$). The path of the bifurcation analysis is along the white horizontal line in Fig. 8 ($\dot{\gamma }=0.9,{\lambda }_{\mathrm{K}}=0.6,\sigma =0$). Solid (dashed) lines indicate stable (unstable) fixpoints. The limit cycles are illustrated by vertical lines between the maximal and minimal values for the projection on the largest eigenvalue $\mu ={\mu }_3$ as the order parameter. Note the bistable region of wagging states [red (dark gray) limit cycles] and kayaking-tumbling states [blue (light gray) limit cycles] between the period doubling bifurcation (PD) and the limit point of cycle (LPC) bifurcation.