Shear-stress-controlled dynamics of nematic complex fluids

Based on a mesoscopic theory we investigate the nonequilibrium dynamics of a sheared nematic liquid, with the control parameter being the shear stress ${\sigma }_{xy}$ (rather than the usual shear rate, $\dot{\gamma }$). To this end we supplement the equations of motion for the orientational order parameters by an equation for $\dot{\gamma }$, which then becomes time dependent. Shearing the system from an isotropic state, the stress-controlled flow properties turn out to be essentially identical to those at fixed $\dot{\gamma }$. Pronounced differences occur when the equilibrium state is nematic. Here, shearing at controlled $\dot{\gamma }$ yields several nonequilibrium transitions between different dynamic states, including chaotic regimes. The corresponding stress-controlled system has only one transition from a regular periodic into a stationary (shear-aligned) state. The position of this transition in the ${\sigma }_{xy}\text{−}\dot{\gamma }$ plane turns out to be tunable by the delay time entering our control scheme for ${\sigma }_{xy}$. Moreover, a sudden change in the control method can stabilize the chaotic states appearing at fixed $\dot{\gamma }$.

Figure 1

(a) Shear stress ${\sigma }_{xy}$ as a function of the shear rate $\Gamma$ (“constitutive curve”) at temperature $\theta =1.75$, corresponding to the isotropic phase of the equilibrium system. Results from calculations at fixed $\Gamma \left({\sigma }_{xy}\right)$ are indicated by open circles (filled triangles). Parts (b) and (c) show the corresponding behavior of the viscosity $\eta$ and the Maier-Saupe order parameter $S$, respectively.

Figure 2

(a) Instantaneous shear rate and (b) shear stress as functions of time under controlled stress conditions $\left({\sigma }_{xy}^{\text{imp}}=0.5\right)$ at $\theta =1.75$. Included are results for four relaxation times $\tau$. The time steps resulting from our adaptive algorithm are $\Delta t\approx 0.3$ for all values of $\tau$ considered.

Figure 3

Same as Fig. 1, but at temperature $\theta =1.25$ close above the $I\text{−}N$ transition of the equilibrium system.

Figure 4

Shear stress ${\sigma }_{xy}$ as a function of the shear rate $\Gamma$ at temperature $\theta =0$, within the nematic phase of the equilibrium system. Results from calculations at fixed $\Gamma \left({\sigma }_{xy}\right)$ are indicated by open (filled) circles. Calculations at fixed stress have been performed at ${\tau }_{\text{g}}=1$. The abbreviations “KT,” “W,” “C,” and “A” indicate the orientational behavior and are explained in the main text. The inset shows the corresponding behavior of the viscosity.

Figure 5

Enlarged version of the stress-strain curve in Fig. 4. The lines indicate the result of constant-stress calculations with different values of the relaxation time ${\tau }_{\text{g}}$.

Figure 6

Instantaneous stress as function of time for different control times ${\tau }_{\text{g}}$. The average shear stress is fixed at ${\sigma }_{xy}^{\text{imp}}=0.575$. The resulting time steps are $\Delta t\approx 0.01$ for all values of $\tau$ considered.

Figure 7

Instantaneous shear rate as function of time for different control times ${\tau }_{\text{g}}$. Parameters as in Fig. 6.

Figure 8

Phase portraits of the in-plane components of the pressure tensor, $a_2$ versus $a_1$, for different control times ${\tau }_{\text{g}}$. The average shear stress is fixed at ${\sigma }_{xy}^{\text{imp}}=0.575$.

Figure 9

Same as Fig. 8, but for the out-of plane components $a_4$ versus $a_3$.

Figure 10

Shear stress and shear rate as functions of time in systems where the control method switches from fixed shear rate $\left(\Gamma =3.7\right)$ to fixed shear stress $\left({\sigma }_{xy}^{\text{imp}}=2.6,{\tau }_{\text{g}}=1\right)$. The switch occurs at (a) $N_{\text{s}}=5\times {10}^4$ time steps, (b) $N_{\text{s}}=1\times {10}^5$ time steps. The resulting time steps are $\Delta t\approx 0.01$ for both values of $N_{\text{s}}$.

Figure 11

Same as Fig. 10, but for the components $a_1$ and $a_2$ of the alignment tensor.