Nonequilibrium steady-state response of a nematic liquid crystal under simple shear flow and electric fields

The effect of a dc electric field on the response of a nematic liquid crystal under shear flow has been investigated by measuring the shear stress response to an ac electric field used as a probe. It was found that both the first- and second-order responses do not vanish at high frequencies, but have constant nonzero values. The experimental results are in good agreement with calculations based on the Ericksen-Leslie theory. The role of the Parodi relation (which is derived from the Onsager reciprocal relation) in the stress response is discussed.

Figure 1

(a) A simple shear flow is applied along the $x$ axis with a velocity gradient parallel to the $z$ axis. An electric field is applied along the $z$ direction. (b) We define $\theta$ to be the angle between the director and the $x$ axis when the electric field is applied.

Figure 2

$E_0$ dependence of ${\theta }_0$ calculated from the EL theory. The angle ${\theta }_0$ increases monotonically and tends to saturate when $E_0>200\mathrm{V}{\mathrm{mm}}^{-1}$.

Figure 3

Experimental setup. Electric fields are applied between the top and bottom plates. For the electro-optical experiment we add a light source and a microscope to observe the change of transmitted intensity.(In the figure: dc and ac electric field source, microscope, sample, rheometer.)

Figure 4

Dependence of $|{\sigma }_{1,1}|$ on $\Delta E$ (a) and $|{\sigma }_{2,2}|$ on $\Delta E^2$ (b) at $\dot{\gamma }=10{\mathrm{s}}^{-1}$ and $E_0=100\mathrm{V}{\mathrm{mm}}^{-1}$. Linear relations are obtained at low ac electric fields.

Figure 5

Frequency dispersion of $\Delta {\sigma }_{1,1}$ as a function of angular frequency $\omega$. The theoretical curves are obtained from Eq. (15). A characteristic nonzero plateau, which is due to the dc electric field, appears at high frequencies in (b)–(d).

Figure 6

$E_0$ dependence of the first-order response at very low frequency ($0.23\mathrm{rad}{\mathrm{s}}^{-1}$ for the experiment and exactly zero for the theory), $\mathrm{Re}\left[\Delta {\sigma }_{1,1}(\omega \to 0)\right]$. Also shown is the dependence of plateau height $\mathrm{Re}\left[\Delta {\sigma }_{1,1}(\omega \to \infty )\right]$. For dc electric fields higher than around $90\mathrm{V}{\mathrm{mm}}^{-1},\mathrm{Re}\left[\Delta {\sigma }_{1,1}(\omega \to \infty )\right]$ becomes larger than $\mathrm{Re}\left[\Delta {\sigma }_{1,1}(\omega \to 0)\right]$.

Figure 7

Frequency dispersion of $\Delta {\sigma }_{2,2}$ as a function of angular frequency $\omega$. The theoretical curves are obtained from Eq. (16). A plateau appears under dc electric fields as well as in the first-order response.

Figure 8

$E_0$ dependence of the first-order response at a very low frequency ($0.23\mathrm{rad}{\mathrm{s}}^{-1}$ for the experiment and exactly zero for the theory), $\mathrm{Re}\left[\Delta {\sigma }_{2,2}(\omega \to 0)\right]$. Also shown is the dependence of plateau height $\mathrm{Re}\left[\Delta {\sigma }_{2,2}(\omega \to \infty )\right]$.

Figure 9

Frequency dispersion of the optical response, which corresponds to the director response, as a function of angular frequency $\omega$. Lines are the experiment and circles are the theory. No plateau is observed, which shows that the plateau observed in the shear stress response should be ascribed to the time derivative of the director but not to $\Delta {\theta }_{1,1}$ itself.